Bergeron–Garsia Science Fiction conjecture for intersections of Garsia–Haiman modules

Let $\nu$ be a partition and let $\mu^{(1)},\dots,\mu^{(k)}\subseteq\nu$ be $k$ distinct partitions such that

$$|\nu/\mu^{(1)}|=\cdots=|\nu/\mu^{(k)}|=1.$$

For each partition $\mu$, let $V_\mu$ be the associated Garsia–Haiman module, let $\widetilde{H}_\mu[X;q,t]$ be the modified Macdonald polynomial, and set

$$T_\mu:=\prod_{(i,j)\in\mu}t^{i-1}q^{j-1}.$$

Science Fiction conjecture. The bigraded $\mathfrak{S}_n$-module $\bigcap_{i=1}^{k}V_{\mu^{(i)}}$ has dimension $\frac{n!}{k}$, and its Frobenius characteristic is

$$\operatorname{Frob}\left(\bigcap_{i=1}^{k}V_{\mu^{(i)}},q,t\right)=\sum_{i=1}^{k}\prod_{j\neq i}\dfrac{T_{\mu^{(j)}}}{T_{\mu^{(j)}}-T_{\mu^{(i)}}}\widetilde{H}_{\mu^{(i)}}[X;q,t].$$

This is presented as an implication of the Bergeron–Garsia Science Fiction conjecture. The first assertion is called the $\frac{n!}{k}$-conjecture, and the source notes that the $\frac{n!}{2}$ case for hook shapes was recently proved; the general statement remains unresolved in the supplied text.