Purity conjecture for the stable cohomology of relative completion

Let ${\mathfrak u}_{g,n+\vec r}$ be the prounipotent Lie algebra of the relative completion, equipped with its weight filtration, and write $\operatorname{Gr}^W_m$ for its weight-$m$ graded piece. Purity conjecture. The weight $m$ stable cohomology of ${\mathfrak u}_{g,n+\vec r}$ is of degree $m$:

$$\operatorname{Gr}^W_m H^{\bullet}({\mathfrak u}_{g,n+\vec r}) = \operatorname{Gr}^W_m H^m({\mathfrak u}_{g,n+\vec r})$$

for all $g\geq m/3$. Purity would allow the computation of $\operatorname{Gr}^W_j{\mathfrak u}_{g,n+\vec r}$ in the representation ring of ${\mathrm{Sp}}(H)$ for $j\leq m$; the conjecture is presented as a consequence expected from stable comparison, but is not established here.