The integrality conjecture for character varieties

Let $g,d\in\mathbb{Z}_{\geq 1}$, let $\Sigma_g$ be a genus-$g$ surface, and let $\operatorname{Rep}_d(\Sigma_g)$ denote the stack of $d$-dimensional representations of its fundamental group. Let $\overline{\operatorname{Rep}}_d^{\operatorname{\zeta}}(\Sigma_g)$ be the smooth quasiprojective variety of twisted representations modulo $\operatorname{PGL}_d$, and let $\mathbb{L}$ denote the Lefschetz mixed Hodge structure. Then the following holds as an isomorphism of $\mathbb{N}$-graded mixed Hodge structures:

The integrality conjecture.

$$\bigoplus_{d\in\mathbb{N}}\operatorname{H}_c\left(\operatorname{Rep}_d(\Sigma_g),\mathbb{Q}\right)\otimes\mathbb{L}^{\otimes(1-g)n^2}\cong \operatorname{Sym}\left(\bigoplus_{d\geq 1}\operatorname{H}_c\left(\overline{\operatorname{Rep}}_d^{\operatorname{\zeta}}(\Sigma_g),\mathbb{Q}\right)\otimes \operatorname{H}_c(\operatorname{pt}/\mathbb{C}^*,\mathbb{Q})\otimes\mathbb{L}^{\otimes(1-g)n^2}\right).$$

This is the integrality conjecture attributed in the source to; it predicts that the compactly supported cohomology of the character stacks is freely generated, in the symmetric-algebra sense, by the cohomology of the twisted character varieties together with the cohomology of $\operatorname{pt}/\mathbb{C}^*$. The formula is known in genus one in the form described immediately before the conjecture, while the general case is presented as conjectural.