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 $-2\pi\ln\tau$.