The five-term case of Goncharov's Depth Conjecture

Let $F$ be the field under consideration, let $k$ be a positive integer, let $x_2,\ldots,x_k\in F^\times$, and let $w_0,\ldots,w_4\in\mathbb{P}^1(F)$. For each $i$, omit $w_i$ from the five-tuple and use the resulting five-term argument in the first slot of the motivic multiple polylogarithm $\operatorname{Li}^{\mathcal{L}}_{k\;1,\ldots,1}$. Five-term depth conjecture. The combination

$$\sum_{i=0}^4(-1)^i\operatorname{Li}^{\mathcal{L}}_{k\;1,\ldots,1}([w_0,\ldots,\widehat{w_i},\ldots,w_4],x_2,\ldots,x_k)$$

has depth at most $k-1$. This is the remaining relation needed in the source's proposed proof of injectivity for the relevant depth-graded coproduct map; its general validity is left open.