Schiffmann's positivity conjecture for Kac polynomials of smooth projective curves

Let $(g,r,d)\in\mathbf{N}\times\mathbf{Z}_{\geqslant 1}\times\mathbf{Z}$ be a triple consisting of a genus $g$, a rank $r$, and a degree $d$. The polynomial $A_{g,r,d}(z_1,\ldots,z_{2g})$ is understood as a character candidate for a representation of $\mathrm{GSp}(2g,\mathbf{C})$. Schiffmann's positivity conjecture. There exists a representation of $\mathrm{GSp}(2g,\mathbf{C})$ whose character is $A_{g,r,d}(z_1,\ldots,z_{2g})$. This strengthens Schiffmann's earlier positivity result; the representation is expected to be given by the relative BPS cohomology of the Dolbeault moduli space over the moduli stack of smooth genus-$g$ curves.