Genus-two relations conjecture for the graded Johnson homomorphism

Let $H$ be the standard symplectic representation in genus $2$, let $V_\boxplus$ be the weight $-2$ ${\mathrm{Sp}}(H)$-module in the minimal presentation

$$\operatorname{Gr}^W_{\bullet} {\mathfrak u}_2 = {\mathbb L}(V_\boxplus)/(R_4,R_6,R_8,R_{10},R_{14}),$$

and, after decomposing a closed genus-$2$ surface into two surfaces of type $(1,\vec{1})$, let ${\mathbf e}_{2j}'$ and ${\mathbf e}_{2k}”$ denote the corresponding elements supported on the two components. Genus-two relations conjecture. When $2n=6,8,10,14$, the relations

$$[{ \mathbf e}_{2j}',{\mathbf e}_{2k}”]=0$$

with $j+k=n$ generate $R_{2n}$ as an ${\mathrm{Sp}}(H)$-module. These relations would provide the conjectural indecomposable relations in the genus-two relative completion; determining why Eisenstein-series weights govern them remains mysterious.