Calabi–Yau fibration conjecture for the stable-sheaf moduli space

Let $X$ be a Calabi–Yau threefold, let $\beta>0$, and let $\operatorname{Sh}_{\beta}(X)$ be the moduli space used to define the Gopakumar–Vafa invariants, equipped with its $d$-critical structure $s$ and Hilbert–Chow morphism

$$\pi:\operatorname{Sh}_{\beta}(X)^{\mathrm{red}}\longrightarrow\operatorname{Chow}_{\beta}(X).$$

A morphism of a $d$-critical scheme to the Chow variety is a CY fibration when the canonical data satisfy the paper's CY-fibration condition. Calabi–Yau fibration conjecture. The $d$-critical scheme $(\operatorname{Sh}_{\beta}(X),s)$ in the paper's theorem is a CY fibration over $\operatorname{Chow}_{\beta}(X)$; consequently, the local invariants $n_{g,\gamma}^{\mathrm{loc}}$ and global invariants $n_{g,\beta}$ are defined. This conjecture is the foundational well-definedness assertion for the proposed invariants; the source gives no resolution.