The divided-differential-operator dimension conjecture for Vandermonde determinants

Let $n$ be a positive integer, let $p$ be a prime with $p^2\geq n$, and let $k$ be any field of characteristic $p$. Let $\triangle_n$ be the Vandermonde determinant, let $\mathbb D_n^k$ be the algebra of divided differential operators in $n$ variables, and let

$$\operatorname{Diff}_n(\triangle_n):=\mathbb C[\partial_{X_1},\dots,\partial_{X_n}]\cdot\triangle_n.$$

Divided-differential-operator dimension conjecture. One has

$$\dim_k\mathbb D_n^k(\triangle_n)=\dim_{\mathbb C}\operatorname{Diff}_n(\triangle_n).$$

The paper motivates this as a characteristic-$p$ analogue of the classical Vandermonde dimension result and notes the restriction $p^2\geq n$. Its resolution is not supplied in the source.