Bergeron–Garsia $n!/k$ intersection conjecture

Let $\lambda$ be an integer partition of $n+1$, and let $\mu^{(1)},\ldots,\mu^{(k)}$ be partitions of $n$, each obtained from $\lambda$ by removing a removable cell from its Young diagram. Let $\mathcal{H}_{\mu^{(i)}}$ denote the corresponding Garsia–Haiman modules. Bergeron–Garsia $n!/k$ conjecture. The intersection of these modules should satisfy

$$\dim\left(\bigcap_{i=1}^k\mathcal{H}_{\mu^{(i)}}\right)=\frac{n!}{k}.$$

The paper studies this conjecture and proves the case $k=2$ for hook shapes, but the displayed assertion is stated in full generality here.