Here arXiv:math/0510287, Golishev proposed the following conjecture:

The hypergeometric pullback conjecture. Let $X$ be a Fano variety. Then,for any constituent $C$ of the quantum D-module $Q$ there exists a torus $T_C$, a morphism of tori $h_C$ : $T_{NS^∨}$ −→ $T_C$ and a hypergeometric D-module $H_C$ on $T_C$ such that $C$ is isomorphic to a constituent of the pullback $h^!H_C$ on some open subset $U$ of $T_{NS^∨}$. ($NS^∨$ is the Neron-Severi dual Torus)

It seems this conjecture is relevant in order to understand the mirror symmetry of Fano varieties. In addition,in the paper, Golyshev points out that the conjecture is also relevant in the problem of classifying the so-called differential equations of type DN (please, see the paper for their definition)

I would like to ask what consequences of the hypergeometric pullback conjecture are known. I would also like to ask whether this conjecture is related with other (known) facts about the mirror symmetry of Fano varieties.

Thanks in advance for your cooperation (and I apologize if the question looks too localized)


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