Point-independence conjecture for log BPS numbers

Let $(S,E)$ be a rational log Calabi-Yau surface with smooth divisor and let $\beta\in\operatorname{H}_2(S,\mathbb{Z})$. For $P\in E(\beta)$, let $m^P_\beta$ denote the log BPS number at $P$, and let $m^{tot}_\beta$ be the total log BPS number; write $w$ for the relevant intersection number. Point-independence conjecture. For all $P,P'\in E(\beta)$,

$$m^P_\beta=m^{P'}_\beta.$$

Equivalently, for all $P\in E(\beta)$,

$$m^{tot}_\beta=w^2m^P_\beta.$$

The claim asserts that the log BPS contribution is independent of the point of maximal tangency. The surrounding text notes that the corresponding log counts can differ before imposing this conjectural BPS-level symmetry.