The dimension conjecture for weighted diagonal harmonics

Let $\mathcal D_n^{\mathbf a}$ be the common polynomial zero space in $\mathbb C[\mathbf{x},\mathbf{y}]$ of the operators

$$\sum_{i=1}^n a_i\,\partial_{x_i}^k\partial_{y_i}^j,$$

for all $k,j\in\mathbb N$ with $k+j>0$. Assume that $\mathbf a=(a_1,\ldots,a_n)$ satisfies

$$\sum_{k\in K}a_k\neq0$$

for every nonempty subset $K\subseteq\{1,\ldots,n\}$. Weighted diagonal-harmonics conjecture. The space $\mathcal D_n^{\mathbf a}$ is bigraded and has dimension $(n+1)^{n-1}$. This extends the corresponding diagonal-harmonics question from equal weights to weights satisfying the stated nonvanishing condition.