Schiffmann's Lie algebra and PBW conjecture for character varieties

Let

$$\mathcal{H}^{\operatorname{\mathtt{Betti}}}_g=\bigoplus_{n\geq 0}\mathrm{H}_c(\mathcal M^{\operatorname{\mathtt{Betti}}}_{g,n},\mathbb{Q})\{(1-g)n^2\}^*,$$

and let $u$ be a formal variable. Schiffmann's conjecture. There is a Lie algebra structure on

$$\bigoplus_{n\geq 1}\mathrm{H}_c(\mathcal M^{\operatorname{\mathtt{Betti}},\operatorname{\mathtt{tw}}}_{g,n},\mathbb{Q})\{(1-g)n^2\}^*,$$

and a filtration $Y$ on $\mathcal{H}^{\operatorname{\mathtt{Betti}}}_g$ such that

$$\operatorname{Gr}_{\bullet}^Y(\mathcal{H}^{\operatorname{\mathtt{Betti}}}_g)\cong\mathcal{U}\left(\bigoplus_{n\geq 1}\mathrm{H}_c(\mathcal M^{\operatorname{\mathtt{Betti}},\operatorname{\mathtt{tw}}}_{g,n},\mathbb{Q})\{(1-g)n^2\}^*[u]\right).$$

The conjecture, suggested by Olivier Schiffmann, would imply the main PBW conjecture and is intended to provide a Lie-theoretic explanation of the character-variety generating series; the source gives no resolution status.