Polarization independence of local BPS invariants

Let $S$ be a Gorenstein surface with ample anticanonical sheaf, let $D$ be a polarization of $S$, and let $n^D_\gamma$ be the local BPS invariant associated with a curve class $\gamma$. Polarization-independence conjecture. The invariant $n^D_\gamma$ is independent of the polarization $D$. Independence of $D$ is needed for the local invariant to be intrinsic to $S$ and the curve class. The authors state that they know of no proof, so the conjecture remains open.