Stable intersection-cohomology conjecture for one-dimensional sheaf moduli spaces

Let $(S,H)$ be a polarized surface, let $\beta$ be an effective divisor, and let $M_{\beta,\chi}$ denote the moduli space of polystable one-dimensional sheaves on $S$ with determinant $\mathcal{O}_S(\beta)$ and Euler characteristic $\chi$. Fix an ample divisor $\beta_0$ and write $\beta=n\beta_0$ with $n\in\mathbb{Z}_{>0}$. Stable intersection-cohomology conjecture. For any given $i\in\mathbb{Z}_{\geq0}$ and $\chi\in\mathbb{Z}$, there exists a constant $N(i,\beta_0,\chi)$ depending on $i$, $\beta_0$, and $\chi$ such that for each integer $k\leq i$,

$$\dim \operatorname{IH}^k(M_{\beta,\chi})=\lim_{m\to\infty}b_k(S^{[m]})$$

when $n\geq N(i,\beta_0,\chi)$, where $\operatorname{IH}^*(-)$ denotes intersection cohomology with $\mathbb{Q}$-coefficients. This proposes stabilization in low degrees for rank-zero sheaf moduli, under positivity along a fixed ample ray; the source presents it as an affirmative analogue of the positive-rank stabilization conjecture, with no external resolution supplied.