The Frobenius structure conjecture: weak Gromov–Witten version

Let $(Y,D)$ be a Fano pair. Let $B(\mathbb{Z})$ denote the integral tropical points, with corresponding generators $\vartheta_q$ of $\operatorname{QH}_{\log}^0(Y,D)$, and let $\langle\cdot\rangle$ be the multilinear point function defined from the logarithmic Gromov–Witten numbers.

Weak Gromov–Witten Frobenius structure conjecture. There exists a product $*$ on $\operatorname{QH}_{\log}^0(Y,D)$ making it a commutative associative $\mathbb{Q}[\operatorname{NE}(Y)]$-algebra with identity $\vartheta_{[Y]}$ such that

$$\langle \vartheta_{q_1},\ldots,\vartheta_{q_s}\rangle=\langle\vartheta_{q_1}*\cdots*\vartheta_{q_s}\rangle$$

for all $s$-tuples $q_1,\ldots,q_s\in B(\mathbb{Z})$ with $s\geq1$.

This conjecture asserts that the logarithmic Gromov–Witten point functions of every Fano pair arise from a single associative commutative product. The parser supplies no evidence that it has been resolved.