Cohomological divisibility conjecture for local BPS invariants

Assume that $S$ is a del Pezzo surface and that $\mathcal{M}_\beta$ is the moduli space of one-dimensional stable sheaves on $S$ with class $\beta$ and holomorphic Euler characteristic $1$. Let $P_t(\mathcal{M}_\beta)$ denote its Poincaré polynomial, and let $w$ be the relevant intersection number. Cohomological divisibility conjecture. The Poincaré polynomial $P_t(\mathcal{M}_\beta)$ has $P_t(\mathbb{P}^{w-1})$ as a factor. Consequently, $n_\beta$ is divisible by $w$.

For line, conic, or nef and big classes of arithmetic genus at most $2$, the paper says this is proved in the cited work except for the class $\beta=-2K_{S_8}$.