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

We prove that the weight 6, depth 3, multiple polylogarithm Li₄, ₁, ₁ ( (xyz) ^-1, x, y), or rather its more natural `divergent' incarnation Li₃;₁, ₁, ₁ (x, y, z), satisfies the 6-fold anharmonic symmetries of the dilogarithm Li₂, 1- and ^-1, in each of x, y and z independently, modulo terms of depth 2. 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.