The quantum cohomology conjecture for crepant resolutions of the anticanonical cone

Let $A=\operatorname{Spec}\bigoplus_{n\geq0}H^0(w\mathbb{P},-nK)$ be the anticanonical affine cone over the weighted projective space $w\mathbb{P}$, and let $\widetilde{A}\to A$ be a crepant resolution. Let $H$ be the hypergeometric differential operator associated with the construction, and let $H^{\mathrm{red}}$ denote its reduced operator. Quantum cohomology conjecture. The equation $H\varphi=0$ is the quantum ordinary differential equation of the small quantum cohomology $QH^\bullet\widetilde{A}$, while $H^{\mathrm{red}}\varphi=0$ is a direct summand of that quantum differential equation. Moreover, $QH^\bullet\widetilde{A}$ is closely related to an appropriate variant of the small quantum orbifold cohomology $QH^\bullet_{\mathrm{orb}}A$. This comparison is suggested by the McKay correspondence and the relationship between the cohomology of a crepant resolution and orbifold cohomology; the supplied text does not state whether the proposed relationship has been proved.