PBW filtration conjecture for the character-variety CoHA

Let $\Sigma_g$ be a genus-$g$ surface, let $\mathcal{H}^{\operatorname{\mathfrak{P}}}_{Q_{\Sigma_g},W_{\Sigma_g}}$ be the associated CoHA, and let $\operatorname{Rep}^{\zeta_m}_m(\Sigma_g)$ denote the twisted character variety. PBW filtration conjecture. There is a filtration $F$ such that

$$\operatorname{Gr}_F\bigl(\mathcal{H}^{\operatorname{\mathfrak{P}}}_{Q_{\Sigma_g},W_{\Sigma_g}}\bigr)\cong\mathcal{U}(\mathfrak{g}[u]),$$

where $\mathfrak{g}$ is a $\mathbb{Z}^{Q_{\Sigma_g,0}}$-graded Lie algebra, and there are isomorphisms in the category of mixed Hodge structures

$$\mathfrak{g}_{(m,m,m,m)}\cong\mathrm{H}_c(\operatorname{Rep}^{\zeta_m}_m(\Sigma_g)/\operatorname{PGL},\mathbb{Q})^*\{(1-g)m^2\}.$$

This predicts a PBW-type description of the CoHA and identifies its Lie algebra generators with cohomology of twisted character varieties. The supplied text gives no resolution status.