Higher Nielsen formulae for motivic multiple polylogarithms

Let $F$ be the field under consideration, let $k$ be a positive integer, and let $(I_i)\in\{0,1\}^k$ specify which arguments are specialized to $1$. Let $\operatorname{Sp}_{\{x_i\to1\mid I_i=1\}}$ denote this specialization, and let $\mathcal{D}_{k-1}\mathcal{L}_{2k}(F)$ be the depth filtration through level $k-1$. Higher Nielsen formulae. For every such tuple $(I_i)$,

$$\operatorname{Sp}_{\{x_i\to1\mid I_i=1\}}\operatorname{Li}^{\mathcal{L}}_{k\;1,\ldots,1}(x_1,\ldots,x_k)\in\mathcal{D}_{k-1}\mathcal{L}_{2k}(F).$$

Equivalently, all these specializations have depth at most $k-1$. The formula generalizes the behavior of the motivic multiple polylogarithm when an argument is specialized to $1$; the source does not report a general proof.