Higher Zagier formulas for multiple polylogarithms

Let $k\geq 2$, let $a_1,\dots,a_k$ be elements for which the displayed multiple polylogarithms are defined, and let $\mathcal{L}_{2k}$ be the weight-$2k$ component of the Lie coalgebra, with depth measured by its depth filtration. Higher Zagier formulas. The elements

$$\operatorname{Li}_{k;1,1,\dots,1}^{\mathcal{L}}(a_1,a_2,\dots,a_k)+\operatorname{Li}_{k;1,1,\dots,1}^{\mathcal{L}}(1-a_1,a_2,\dots,a_k)$$

and

$$\operatorname{Li}_{k;1,1,\dots,1}^{\mathcal{L}}(a_1,a_2,\dots,a_k)+\operatorname{Li}_{k;1,1,\dots,1}^{\mathcal{L}}\left(\frac{1}{a_1},a_2,\dots,a_k\right)$$

belong to $\mathcal{L}_{2k}$ and have depth at most $k-1$. These formulas generalize Zagier's identities and Gangl's weight-four formula. They are proved for $k=2$ and remain open for $k\geq 3$.