Complex product expansion conjecture for Calabi–Yau threefolds

Let $X$ be a complex projective Calabi–Yau threefold. For each homology class $\beta$ and genus $g\geq 0$, let $e_{\beta,g}(X)$ be an integer, and define

$$f_g(t)=\sum_{m\geq 0}x_{g,m,0}t^m.$$

Complex product expansion conjecture. The element $(X,t^{[\cdot]})\in H^0_c(\mathcal Z^{\mathrm{semi\textrm{-}Fano}}(\operatorname{Cpx}_3))[[t^{H_2(X)}]]$ is an infinite product

$$(X,t^{[\cdot]})=\prod_\beta\prod_{g\geq 0}f_g(t^\beta)^{e_{\beta,g}(X)},$$

for unique integer invariants $e_{\beta,g}(X)\in\mathbb Z$. This seeks a complex-geometric analogue of the Ionel–Parker product expansion; the existence and uniqueness of these invariants are conjectural in the stated setting.