  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 $d=2$.
