Cao–Maulik–Toda Katz/GV conjecture for one-dimensional sheaves on Calabi–Yau fourfolds

Let $X$ be a Calabi–Yau $4$-fold and let $M_1(X,\beta)$ be the moduli scheme of one-dimensional stable sheaves $F$ with $[F]=\beta\in H_2(X,\mathbb Z)$ and $\chi(F)=1$. Let $[M_1(X,\beta)]^{\mathrm{vir}}$ be its virtual class. For $\gamma\in H^4(X,\mathbb Z)$, define

$$\tau(\gamma)=(\pi_M)_*(\pi_X^*\gamma\cup\operatorname{ch}_3(\mathbb F)),$$

where $\mathbb F$ is the universal sheaf and $\pi_X,\pi_M$ are the projections from $X\times M_1(X,\beta)$. Cao–Maulik–Toda Katz/GV conjecture. For a suitable choice of orientations,

$$\int_{[M_1(X,\beta)]^{\mathrm{vir}}}\tau(\gamma)=n_{0,\beta}(\gamma),$$

where $n_{0,\beta}(\gamma)$ is the genus $0$ Gopakumar–Vafa type invariant.

This is proposed as a sheaf-theoretic analogue of the Katz/GV conjecture for Calabi–Yau fourfolds. The source gives examples and comparisons supporting it, but states it as conjectural.