The quantum rim-hook rule for cominuscule homogeneous spaces

Let $X=G/P$ be a cominuscule homogeneous space, let $\ideal\subideal\minposet$ be a generalized Young diagram, and let $\sigma_\ideal$ denote the corresponding Schubert class. Let $\idealPPrime\subideal\minposet$ be the generalized rim-hook associated with the cominuscule type, and let $\sigma_{\ideal^-}$ denote the Schubert class of the complement of $\idealPPrime$ in $\ideal$ when $\idealPPrime\subset\ideal$, and $0$ otherwise. The sum below runs over all generalized Young diagrams $\ideal^+\subideal\minposet$ obtained by adding one box to $\ideal$. Quantum rim-hook conjecture. For any cominuscule homogeneous space, the quantum product with the hyperplane class satisfies

$$\sigma_\ideal *_q \sigma_{\yng(1)}=q\sigma_{\ideal^-}+\sum_{\ideal^+}\sigma_{\ideal^+}.$$

This generalizes the rim-hook rule for Grassmannians to arbitrary cominuscule homogeneous spaces. The preceding discussion notes that it agrees with the classical derivation formula at $q=0$ and is suggested by the quantum term and the cited results, but the supplied text gives no resolution of the proposed generalization.