Toda's chi-independence conjecture for BPS perverse sheaves

Let $S$ be a smooth projective surface, let $\alpha\in\mathrm{H}^2(S,\mathbb{Z})$ be a curve class, let $\delta\in\mathrm{H}^4(S,\mathbb{Z})$ be the class of a point, and set $\nu=\alpha+n\delta$. Let $B_S(\alpha)$ be the Chow variety parametrizing algebraic one-cycles on $S$ of class $\alpha$, let $M_S(\nu)$ be the corresponding moduli space, and let $\pi_S\colon M_S(\nu)\to B_S(\alpha)$ be the support map. For its 2D BPS perverse sheaf, write $\mathcal{BPS}_{M_S(\nu)}$. For any $\gamma\in B_S(\alpha)$, define

$$\Phi(\alpha+n\delta)=\sum_{i\in\mathbb{Z}}\chi\big({}^{{\operatorname{p}}}\mathcal{H}^i((\pi_S)_*\mathcal{BPS}_{M_S(\alpha+n\delta)})|_{\gamma}\big)\cdot y^i\in\mathbb{Z}[y^{\pm1}].$$

Toda's chi-independence conjecture. For any $\gamma\in B_S(\alpha)$, the invariant $\Phi(\alpha+n\delta)$ is independent of $n\in\mathbb{Z}$.

The conjecture tests whether Gopakumar--Vafa invariants depend only on the underlying curve class, rather than on the choice of Euler-characteristic parameter $n$. Its Euler-characteristic specialization at $y=-1$ has been proved by Maulik and Thomas, while the full $y$-refined independence is not established here.