Quantum D-module conjecture for primitive extremal transitions

Let $X$ and $Y$ be smooth projective varieties related by a primitive extremal transition. Denote by $\mathcal{H}(X)$ the ambient part of the quantum $D$-module of $X$. After analytic continuation of $\mathcal{H}(X)$ over the extended Kähler moduli, write the resulting $D$-module as $\bar{\mathcal{H}}(X)$. Then there are a divisor $E$ and a submodule $\bar{\mathcal{H}}^E(X)\subseteq\bar{\mathcal{H}}(X)$ with maximum trivial $E$-monodromy such that

$$\bar{\mathcal{H}}^E(X)|_E\simeq\mathcal{H}(Y),$$

where $\bar{\mathcal{H}}^E(X)|_E$ denotes the restriction to $E$. This conjecture proposes a relationship between the ambient quantum $D$-modules of the two varieties across a primitive extremal transition, generalizing the proposed comparison for cubic extremal transitions; the statement concerns analytic continuation and restriction rather than equivalence of the full Gromov–Witten theories.