Symmetries of weight 6 multiple polylogarithms and Goncharov's Depth Conjecture

We prove that the weight 6, depth 3, multiple polylogarithm $\mathrm{Li}_{4,1,1}((xyz)^{-1}, x, y)$, or rather its more natural `divergent' incarnation $\mathrm{Li}_{3;1,1,1}(x,y,z)$, satisfies the 6-fold anharmonic symmetries of the dilogarithm $\mathrm{Li}_2$, $\lambda \mapsto 1-\lambda$ and $\lambda \mapsto \lambda^{-1}$, in each of $x$, $y$ and $z$ independently, modulo terms of depth $\leq2$. This establishes the `higher Zagier' part of the weight 6, depth 3, reduction conjectured by Matveiakin and Rudenko. Together with their proof of the `higher Gangl' part of the weight 6, depth 3, reduction (which is formulated modulo the `higher Zagier' part), we establish Goncharov's Depth Conjecture in the case of weight 6, depth 3.