Free-algebra conjecture for the Betti character-stack Hall algebra

Let $g\geq 2$, let $\Sigma_g$ be a genus-$g$ surface, and let $p\colon \mathfrak{M}_{g,r}^{\operatorname{Betti}}\to \mathcal{M}_{g,r}^{\operatorname{Betti}}$ be the semisimplification morphism for $r$-dimensional representations of the surface-group algebra. Let $\operatorname{\mathcal{RA}}_{\Sigma_g}$ be the relative Hall algebra object and let $\mathcal{H}_{\Sigma_g}$ be the associated Borel–Moore homology Hall algebra. Free-algebra conjecture. There are isomorphisms

$$\uptau_{\leq 0}\operatorname{\mathcal{RA}}_{\Sigma_g}\cong \operatorname{\mathbf{Free}}_{\oplus}\left(\bigoplus_{r\geq 1}\underline{\operatorname{\mathcal{IC}}}_{\mathcal{M}_{g,r}^{\operatorname{Betti}}}\right)$$

and

$$\mathcal{H}_{\Sigma_g}\cong \operatorname{\mathbf{Sym}}\left(\operatorname{\mathbf{Lie}}\left(\bigoplus_{r\geq 1}\underline{\operatorname{\mathcal{IC}}}_{\mathcal{M}_{g,r}^{\operatorname{Betti}}}\right)[u]\right).$$

The conjecture predicts a free-algebra structure for the truncated relative Hall algebra and the corresponding symmetric-algebra description of its homology; the source does not state that these isomorphisms are known.