Show that $\left(\frac{x_1^{x_2}}{x_2}\right)^p+\left(\frac{x_2^{x_3}}{x_3}\right)^p+\cdots+\left(\frac{x_n^{x_1}}{x_1}\right)^p\ge n$ for any $p\ge1$ The inequality $\sqrt{\frac{a^b}{b}}+\sqrt{\frac{b^a}{a}}\ge 2$ for all $a,b>0$ was shown here using first-order Padé approximants on each exponent, where the minimum is attained at $a=b=1$.
By empirical evidence, it appears that inequalities of this type hold for an arbitrary number of variables. We can phrase the generalised problem as follows.

Let $(x_i)_{1\le i\le n}$ be a sequence of positive real numbers. Define $\boldsymbol a=\begin{pmatrix}a_1&\cdots&a_n\end{pmatrix}$ such that $a_k=x_k^{x_{k+1}}/x_{k+1}$ for each $1\le k<n$ and $a_n=x_n^{x_1}/x_1$. How do we show that $$\|\boldsymbol a\|_p^p\ge n$$ for any $p\ge1$?

As before, AM-GM is far too weak since the inequality $\displaystyle\|\boldsymbol a\|_p^p\ge 2\left(\prod_{\text{cyc}}\frac{x_1^{x_2}}{x_2}\right)^{1/{2p}}$ does not guarantee the result when at least one $x_i$ is smaller than $1$. We can eliminate the exponent on the denominator by taking $x_i=X_i^{1/p}$ so that $\displaystyle\|\boldsymbol a\|_p^p=\sum_{\text{cyc}}\frac{X_1^{X_2^{1/p}}}{X_2}$ but the approximant approach no longer becomes feasible; even in the case where $p$ is an integer the problem reduces to a posynomial inequality of rational degrees. Perhaps there are some obscure $L^p$-norm/Hölder-type identities of use but I'm at a loss in terms of finding references.
Empirical results: In the interval $p\in[1,\infty)$, Wolfram suggests that the minimum is $n$ (Notebook result) which is obtained when $\boldsymbol a$ is the vector of ones. However, we note that in the interval $p\in(0,1)$, the empirical minimum no longer displays this consistent behaviour as can be seen in this Notebook result.
The sequence $\approx(1.00,2.00,2.01,3.36,3.00,4.00)$ appears to increase almost linearly every two values, but I cannot verify it for a larger number of variables due to instability in the working precision.
 A: We prove the result by first deriving a useful inequality for the function $(x,y)\mapsto (x^y/y)^p$.

It is clear that $$(x^y/y)^p \ge \left(\min_{\alpha \in \mathbb{R}_+} (\alpha x)^{\alpha y}/(\alpha y)\right)^p.\tag{1}$$ For fixed $x$ and $y$, we can find the minimum of the function $\alpha\mapsto (\alpha x)^{\alpha y}/(\alpha y)$  by taking derivatives and setting to $0$. Doing this (e.g., in Mathematica) shows that the minimum is achieved by $\alpha^\star = 1/({yW(ex/y)})$, where $W$ is the Lambert W function. Plugged into $(1)$, this gives
$$(x^y/y)^p \ge \left[\left(\frac{x/y}{W(ex/y)}\right)^{1/W(e x/y)}W(ex/y)\right]^p$$
Note that the right hand side is a function of the ratio $x/y$ only. It will be convenient to write it as a function of the logarithm of $x/y$,
$(x^y/y)^p\ge f(\ln x/y)$
where $$f(a):=\left[\left(\frac{e^a}{W(e^{a+1})}\right)^{1/W(e^{a+1})} W(e^{a+1})\right]^p.$$
We will use that $f(a)$ is a convex function for all $a\in\mathbb{R}$ and $p\ge 1$, since its second derivative is non-negative. This can be verified in Mathematica by running
FullSimplify[D[((Exp[a]/ProductLog[Exp[a + 1]])^(1/ProductLog[Exp[a + 1]])*ProductLog[Exp[a + 1]])^p, {a, 2}], Assumptions -> {Element[a, Reals]}]

which gives
$$f''(a) = \frac{p e^{p-\frac{p}{W\left(e^{a+1}\right)}} W\left(e^{a+1}\right)^{p-2} \left((p-1) W\left(e^{a+1}\right)+p\right)}{W\left(e^{a+1}\right)+1}$$
which is non-negative for $p\ge 1$.

We are now ready to prove the result:
$$
\sum_{i=1}^{n} \left(\frac{x_{i}^{x_{i+1}}}{x_{i+1}}\right)^p \ge n,\tag{2}
$$
where we use the notation $x_{n+1}=x_1$. To begin, write
$$\sum_i \left(\frac{x_{i}^{x_{i+1}}}{x_{i+1}}\right)^p \ge \sum_i f\left(\ln \frac{x_{i}}{x_{i+1}}\right) = n \frac{1}{n} \sum_i  f\left(\ln \frac{x_{i}}{x_{i+1}}\right) \ge n f\left(\frac{1}{n} \sum_i  \ln \frac{x_{i}}{x_{i+1}}\right)\tag{3}$$
where the first inequality uses $(x^y/y)^p\ge f(\ln x/y)$ while the second inequality is Jensen's.
Note that $\sum_i  \ln \frac{x_{i}}{x_{i+1}}=0$. It is also easy to verify from inspection that $f(0)=1$. Plugging into $(3)$ gives
$$n f\left(\frac{1}{n} \sum_i  \ln \frac{x_{i}}{x_{i+1}}\right)= n f(0) = n.$$
Combining with $(3)$ gives the desired result, $(2)$.
A: @Artemy's proof involves a complicated part of proving $f''(a)\ge 0$.
Here is a proof which is a simplification of @Artemy's proof (one can do it by hand; also avoiding the use of the Lambert W function).
Perhaps this is also helpful for the case when $p < 1$ (e.g. $\sqrt{\frac{x^y}{y}}+\sqrt{\frac{y^x}{x}}\ge 2$).

Fact 1: For any given $x, y > 0$, there exists
a unique $u > 0$ such that $\frac{x}{y} = u\mathrm{e}^{u - 1}$. Furthermore,
$$\frac{x^y}{y} \ge u\mathrm{e}^{1 - 1/u}.$$
(The proof is given at the end.)
Using Fact 1, let
$$\frac{x_i}{x_{i + 1}} = u_i\mathrm{e}^{u_i - 1}, \,\, i = 1, 2, \cdots, n - 1; \qquad
\frac{x_n}{x_1} = u_n\mathrm{e}^{u_n - 1}$$
where $u_1, \cdots, u_n > 0$. We have
\begin{align*}
 1 &= \frac{x_1}{x_2}\, \frac{x_2}{x_3} \cdots \frac{x_n}{x_1}\\
 &= u_1 u_2 \cdots u_n \mathrm{e}^{u_1 + u_2 + \cdots + u_n - n}\\
 &\le \left(\frac{u_1 + u_2 + \cdots + u_n}{n}\right)^n \mathrm{e}^{u_1 + u_2 + \cdots + u_n - n}
\end{align*}
which results in
$$\frac{u_1 + u_2 + \cdots + u_n}{n} \ge 1. \tag{1}$$
Let
$$f(u) = (u\mathrm{e}^{1 - 1/u})^p.$$
$f(u)$ is convex on $u > 0$ since
$f''(u) = (u\mathrm{e}^{1 - 1/u})^p
pu^{-4}[(p - 1)u^2 + 2(p - 1)u + p] > 0$ for all $u > 0$.
Also, $f(u)$ is strictly increasing on $u > 0$
since $f'(u) = (u\mathrm{e}^{1 - 1/u})^ppu^{-2}(u + 1) > 0$ for all $u > 0$.
Using Fact 1 and (1), using the convexity and monotonicity of $f(u)$, we have
\begin{align*}
 &\left(\frac{x_1^{x_2}}{x_2}\right)^p
 + \left(\frac{x_2^{x_3}}{x_3}\right)^p
 + \cdots + \left(\frac{x_n^{x_1}}{x_1}\right)^p\\
 \ge\,& f(u_1) + f(u_2) + \cdots + f(u_n)\\
 \ge\,& n\, f\left(\frac{u_1 + u_2 + \cdots + u_n}{n}\right) \\
 \ge\,& n f(1)\\
 \ge\,& n.
\end{align*}
We are done.

Proof of Fact 1:
Since $u\mapsto u\mathrm{e}^{u - 1}$ is strictly increasing on $u > 0$, clearly, there exists a unique $u > 0$ such that $\frac{x}{y} = u\mathrm{e}^{u - 1}$.
We have
$$\ln x = \ln y + \ln u + u - 1. $$
We need to prove that
$$y\ln x - \ln y \ge \ln u + 1 - \frac{1}{u}.$$
It suffices to prove that
$$y(\ln y + \ln u + u - 1) - \ln y \ge \ln u + 1 - \frac{1}{u}.$$
Let
$$F(y) = y(\ln y + \ln u + u - 1) - \ln y - \left(\ln u + 1 - \frac{1}{u}\right).$$
We have
$$F'(y) = \ln y + \ln u + u - \frac{1}{y},$$
and
$$F''(y) = \frac{1}{y} + \frac{1}{y^2} > 0.$$
Also, $F(1/u) = 0$ and $F'(1/u) = 0$.
Thus, $F(y) \ge F(1/u) = 0$ for all $y > 0$.
We are done.
A: Proof for $p ≥ 1$
Since $u^p - 1 ≥ p(u - 1)$ for all $u ≥ 0$, it suffices to prove the result for $p = 1$. That follows from
$$\frac{x^y}{y} - 1 ≥ \frac{1 + y \ln x}{y} - 1
= \ln x + \frac1y - 1 ≥ \ln x + \ln \frac1y = \ln x - \ln y$$
by cyclic summation over $(x, y) = (x_i, x_{i + 1})$.
Conjectured proof for $p ≥ \frac12$
Since $u^p - 1 ≥ 2p(u^{\frac12} - 1)$ for all $u ≥ 0$, it suffices to prove the result for $p = \frac12$. Numerical evidence suggests that
$$\left(\frac{x^y}{y}\right)^{\frac12} - 1 ≥ \frac{\ln x}{2\sqrt[4]{1 + \frac13 \ln^2 x}} - \frac{\ln y}{2\sqrt[4]{1 + \frac13 \ln^2 y}}$$
for all $x, y > 0$. If this is true, cyclic summation yields the desired result.
Counterexample for $0 < p < \frac12$
Let $g(x) = \left(\frac{x^{1/x}}{1/x}\right)^p + \left(\frac{(1/x)^x}{x}\right)^p$. Then $g(1) = 2$, $g'(1) = 0$, and $g''(1) = 4p(2p - 1) < 0$, so we have $g(x) < 2$ for $x$ in some neighborhood of $1$. This yields counterexamples for all even $n$:
$$\left(x, \frac1x, x, \frac1x, \dotsc, x, \frac1x\right), \quad x ≈ 1, \quad 0 < p < \frac12.$$
For $n = 3$, the best counterexample seems to be
$$(0.41398215, 0.73186577, 4.77292996), \quad 0 < p < 0.39158477.$$
A: Partial answer Hint :
In the same vein as Anders Kaseorg we have :
Let $a,x>1$ or $a\geq 1\geq x$ or $a\leq 1\leq x$ or $0.1\leq a\leq 1$ and $0<x\leq 1$ :
$$\sqrt{\frac{a^x}{x}} - \frac{a-1}{a+1} + \frac{x-1}{x+1}-1\geq 0 $$
A sketch of proof can be found here Problem with an inequality : $\sqrt{\frac{a^x}{x}} - \frac{a-1}{a+1} + \frac{x-1}{x+1}-1\geq 0 \,?$
To be continued !
Edit 31/01/2022 :
Another case for $0<c\leq a \leq 1$ and $x>0$ then we have :
$$f\left(x\right)=\sqrt{\frac{a^{x}}{x}}+\sqrt{\frac{x^{c}}{c}}-\frac{\left(a-1\right)}{a+1}+\frac{\left(c-1\right)}{c+1}-2\geq 0$$
