Module-intersection conjecture for Macdonald piece polynomials

Let $S\in\binom{[n]}{k}$ and let $\lambda$ be a partition inside $R(n,k)$. Write $S(\lambda)$ for the collection of $k$-subsets defined by the containment conditions in the paper, and let $R_{\mu^{S'}}$ denote the corresponding Garsia--Haiman module. Let $\widetilde{H}_{\mu^S}^{\lambda}$ be the symmetric function defined using the operator $\pi_{\lambda,S}$ and the specialization $\Phi$. Module-intersection conjecture. One has

$$\widetilde{H}_{\mu^S}^{\lambda}=\operatorname{grFrob}\left(\bigcap_{S'\in S(\lambda)}R_{\mu^{S'}}\right).$$

This gives the conjectural representation-theoretic interpretation underlying the extension of the science fiction conjecture. Its status is not resolved in the supplied text.