Evaluation of the third logarithmic Mahler measure of $1+x+y$

Let

$$\mu_3(1+x+y)$$

denote the third logarithmic Mahler measure of the polynomial $1+x+y$, and let $\operatorname{Ls}_4$ and $\operatorname{Cl}_s$ denote the log-sine and Clausen functions, respectively. Then the evaluation conjecture.

$$\mu_3(1+x+y)=\frac{6}{\pi}\operatorname{Ls}_4\left(\frac{2\pi}{3}\right)-\frac{9}{\pi}\operatorname{Cl}_4\left(\frac{\pi}{3}\right)-\frac{\pi}{4}\operatorname{Cl}_2\left(\frac{\pi}{3}\right)-\frac{13}{2}\zeta(3).$$

The identity is presented as a conjectural closed form suggested by numerical integer-relation computations and related evaluations; the source does not establish whether it has since been proved or disproved.