The $n!$ conjecture for derivative spaces of lattice determinants

Let $\mu$ be a partition of $n$, let $D(\mu)$ be its diagram, and let

$$\Delta_{\mu}=\det\left(x_i^{p_j}y_i^{q_j}\right)_{1\leq i,j\leq n},$$

where $(p_j,q_j)$ are the elements of $D(\mu)$. Define

$$D_{\mu}=\mathbb{C}[\partial\mathbf{x},\partial\mathbf{y}]\Delta_{\mu}$$

to be the space spanned by all partial derivatives of $\Delta_{\mu}$.

$n!$ conjecture. The dimension of $D_{\mu}$ is equal to $n!$.

The conjecture was proposed by Garsia and the author and is the geometric statement underlying the $n!$ theorem and related Macdonald-positivity results; in the source it is subsequently proved.