Stabilization conjecture for refined BPS numbers of one-dimensional sheaf moduli

Let $S$ be a del Pezzo surface, let $\beta$ be an effective divisor, and let $M_{\beta,\chi}$ be the moduli space of one-dimensional sheaves with determinant $\mathcal{O}_S(\beta)$ and Euler characteristic $\chi$. Define the refined BPS numbers by

$$n_\beta^{i,j}:=\dim \operatorname{Gr}^P_i\operatorname{IH}^{i+j}(M_{\beta,\chi}).$$

Refined BPS stabilization conjecture. The refined BPS number $n_\beta^{i,j}$ stabilizes when $\beta$ is sufficiently positive. This conjectures asymptotic stability of the refined BPS invariants as the curve class becomes positive; the source gives partial calculations and states no resolution.