Coskun–Woolf stabilization conjecture for Betti numbers of sheaf moduli spaces

Let $X$ be a smooth projective surface and let $H$ be an ample line bundle. Fix a rank $r>0$ and a first Chern class $c$, and let $M_{X,H}(r,c,\Delta)$ be the corresponding moduli space of sheaves. Coskun–Woolf conjecture. The $i$th Betti number of $M_{X,H}(r,c,\Delta)$ stabilizes to a constant $b_{i,\operatorname{Stab}}(X)$, independent of $r$, $c$ and $H$, as $\Delta$ tends to infinity. This is a conjecture about the expected stabilization of the topology of moduli spaces of sheaves; it is wide open in general, especially for surfaces of general type.