Dilogarithm identity containing the tribonacci constant The motivation of this question is the brilliant conjecture by @Tito Piezas III. In $(4)$ of his question the equation seems to be true for all $n > 1$ real numbers. The case $n=2$ leads us to a dilogarithm identity, where all the dilogarithm terms are known constants. After that I've tried to prove the case $n=3$.
Let $\tau$ denotes the tribonacci constant, defined as the following.
$$\tau = \frac{1}{3}\left(1+\sqrt[3]{19-3\sqrt{33}}+\sqrt[3]{19+3\sqrt{33}}\right) \approx 1.83928675521416\dots$$
Note that $\tau^3-\tau^2-\tau-1=0$.

How could we prove the following conjectured dilogarithm identity?

$$\operatorname{Li}_2\left(\frac{1}{\tau^3}\right)+\operatorname{Li}_2\left(\tau^2\right)+\operatorname{Li}_2\left(\frac{\tau}{\tau+1}\right)$$ 
$$ \stackrel{?}{=}
\frac{1}{2}\ln(1-\tau)\ln\tau - \frac{1}{2}\ln\left(\frac{1}{\tau+1}\right)\ln\left(\frac{\tau}{\tau+1}\right)-\frac{1}{4}\ln(\tau^2)\ln(1-\tau^2)-\ln^2(-\tau)-\frac{7\pi^2}{12},
$$
where $\operatorname{Li}_2$ is the dilogarithm function.
It is quite easy to show that the imaginary part of the sum of dilogarithms is $-2\pi\ln\tau$. 
 A: To simplify the formulas, I will write $\sum_i\operatorname{Li}_2(x_i)\simeq \sum_j\operatorname{Li}_2(y_j)$ if both sides differ by elementary functions of $x_i$, $y_j$. We have in particular
\begin{gather*}
\operatorname{Li}_2(-x)\simeq -\operatorname{Li}_2(x)+\frac12\operatorname{Li}_2\left( x^2\right),\\
\operatorname{Li}_2(x)\simeq -\operatorname{Li}_2\left( x^{-1}\right)\simeq
\operatorname{Li}_2\left( \frac1{1-x}\right),
\end{gather*}
so that 
\begin{gather*}
\operatorname{Li}_2\left( \frac{\tau}{\tau+1}\right)\simeq \operatorname{Li}_2\left( -\tau^{-1}\right)\simeq -\operatorname{Li}_2\left( \tau^{-1}\right)+\frac12\operatorname{Li}_2\left( \tau^{-2}\right),\\
\operatorname{Li}_2\left( \tau^{2}\right)=-\operatorname{Li}_2\left( \tau^{-2}\right).
\end{gather*}
The identity we want to prove can therefore be written as
$$\tag{$\clubsuit$}\operatorname{Li}_2\left( \sigma^{3}\right)-\frac12 \operatorname{Li}_2\left( \sigma^{2}\right)-\operatorname{Li}_2\left( \sigma\right)\simeq0,$$
where $\sigma=\tau^{-1}$ satisfies the algebraic equation $\sigma^3+\sigma^2+\sigma=1$.
Identities of the form $\sum_{k=1}^N r_k\operatorname{Li}_2\left(\alpha^k\right)\simeq0$ with $r_k\in\mathbb Q$ and algebraic $\alpha$ are called dilogarithmic ladders. There exists a well-developed technology of their discovery and many classification results. A classical reference is the book Structural properties of polylogarithms (ed. L. Lewin). In fact, ($\clubsuit$) is a linear combination of ladders (3.80) and (3.81) given on its page 41.
