This is an outline of a proof.
It suffices to prove that
$$a^2b^2c^2d^2(ab+bc+cd+da+ac+bd)\Big[\Big(\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\Big)^2
- \frac{24(a^2+b^2+c^2+d^2)}{ab+bc+cd+da+ac+bd}\Big] \ge 0.$$
Denote LHS by $f(a,b,c,d)$ which is a homogeneous polynomial.
WLOG, let $d = \min(a,b,c,d)$. We use the Buffalo Way.
Let $c = d + s, \ b = d + t, \ a = d + r; \ s, t, r \ge 0.$
We have
$$f(d+r, d+t, d+s, d) = a_8d^8 + a_7d^7 + a_6d^6 + a_5d^5 + a_4d^4 + a_3d^3 + a_2 d^2 + a_1d + a_0$$
where
\begin{align}
a_8 &= 24 r^2+16 r s-32 r t+24 s^2-32 s t+24 t^2, \\
a_7 &= 24 r^3+104 r^2 s-16 r^2 t+104 r s^2-144 r s t+32 r t^2+24 s^3+32 s^2 t-16 s t^2+24 t^3, \\
a_6 &= 6 r^4+96 r^3 s-12 r^3 t+220 r^2 s^2-60 r^2 s t+50 r^2 t^2+96 r s^3+60 r s^2 t-116 r s t^2+60 r t^3+6 s^4+60 s^3 t+50 s^2 t^2-12 s t^3+6 t^4, \\
a_5 &= 3 r^5+23 r^4 s-19 r^4 t+162 r^3 s^2-8 r^3 s t+23 r^3 t^2+162 r^2 s^3+174 r^2 s^2 t-51 r^2 s t^2+79 r^2 t^3+23 r s^4+168 r s^3 t+45 r s^2 t^2-34 r s t^3+17 r t^4+3 s^5+17 s^4 t+79 s^3 t^2+23 s^2 t^3-19 s t^4+3 t^5,\\
a_4 &= 7 r^5 s+r^5 t+34 r^4 s^2-37 r^4 s t-12 r^4 t^2+106 r^3 s^3+122 r^3 s^2 t+r^3 s t^2+43 r^3 t^3+34 r^2 s^4+234 r^2 s^3 t+60 r^2 s^2 t^2+29 r^2 s t^3+20 r^2 t^4+7 r s^5+49 r s^4 t+157 r s^3 t^2+33 r s^2 t^3-39 r s t^4+7 r t^5+7 s^5 t+20 s^4 t^2+43 s^3 t^3-12 s^2 t^4+s t^5,\\
a_3 &= 5 r^5 s^2+2 r^5 s t+19 r^4 s^3-5 r^4 s^2 t-30 r^4 s t^2+2 r^4 t^3+19 r^3 s^4+136 r^3 s^3 t+34 r^3 s^2 t^2+48 r^3 s t^3+9 r^3 t^4+5 r^2 s^5+59 r^2 s^4 t+130 r^2 s^3 t^2+38 r^2 s^2 t^3-15 r^2 s t^4+5 r^2 t^5+16 r s^5 t+43 r s^4 t^2+80 r s^3 t^3-24 r s^2 t^4+2 r s t^5+5 s^5 t^2+9 s^4 t^3+2 s^3 t^4,\\
a_2 &= r^5 s^3+r^5 s^2 t+2 r^4 s^4+15 r^4 s^3 t-16 r^4 s^2 t^2+2 r^4 s t^3+r^3 s^5+29 r^3 s^4 t+50 r^3 s^3 t^2+24 r^3 s^2 t^3+5 r^3 s t^4+r^3 t^5+11 r^2 s^5 t+34 r^2 s^4 t^2+44 r^2 s^3 t^3-10 r^2 s^2 t^4+r^2 s t^5+11 r s^5 t^2+17 r s^4 t^3+4 r s^3 t^4+s^5 t^3, \\
a_1 &= 2 r^4 s^4 t+2 r^4 s^3 t^2+2 r^3 s^5 t+11 r^3 s^4 t^2+6 r^3 s^3 t^3+2 r^3 s^2 t^4+7 r^2 s^5 t^2+9 r^2 s^4 t^3+2 r^2 s^3 t^4+2 r s^5 t^3, \\
a_0 &= r^3 s^5 t^2+r^3 s^4 t^3+r^2 s^5 t^3.
\end{align}
It suffices to prove that $a_i\ge 0$ for $i=0, 1, \cdots, 8.$
Maybe someone can give a nice proof that $a_i\ge 0$ for $i=0, 1, \cdots, 8.$
