Maulik–Toda conjecture for stable-pair and sheaf-theoretic invariants

Let $X$ be a projective Calabi–Yau threefold, let $\beta\in H_2(X,\mathbb Z)$ be a curve class, let $M_{\beta,1}$ be the moduli space of stable one-dimensional sheaves with class $\operatorname{ch}_2(F)=\beta$ and Euler characteristic $1$, and let $\rho:M_{\beta,1}\to\operatorname{Chow}_{\beta}$ be the Chow morphism. Let $\Phi$ be the perverse sheaf associated with Calabi–Yau orientation data, and let $\mathsf{GV}^{\mathrm{unref}}_{\beta}(p)$ denote the unrefined Gopakumar–Vafa polynomial.

Maulik–Toda conjecture. One has

$$\mathsf{GV}^{\mathrm{unref}}_{\beta}(p)=\sum_i\chi\bigl({^p\mathcal H}^{i}(R\rho_*\Phi)\bigr)(-p)^i.$$

Maulik and Toda proposed this equality as the sheaf-theoretic description of the unrefined Gopakumar–Vafa polynomial in terms of perverse cohomology. The parser supplies no resolution evidence, so its status remains open here.