On the Goncharov Depth Conjecture and polylogarithms of depth two

We prove the surjectivity part of Goncharov's depth conjecture. We also show that the depth conjecture implies that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function $\mathrm{Li}_{n-d+1,1,\dots,1}(a_1,a_2,\dots,a_d)$, and we prove this latter statement for $d=2$.