Generalized Haiman ideal Hilbert-series conjecture

Assume $d_1\leq\cdots\leq d_n$, set $d_{ij}=\min(d_i,d_j)$, and define

$$J'(d_1,\ldots,d_n)=\bigcap_{i\neq j}(t_i-t_j,x_i-x_j)^{d_{ij}}\subseteq\mathbb{C}[t_1,\ldots,t_n,x_1,\ldots,x_n].$$

Let $\mathfrak m=(t_1,\ldots,t_n,x_1,\ldots,x_n)$. Generalized Haiman ideal Hilbert-series conjecture. The Hilbert series of $J'(d_1,\ldots,d_n)$ is $H(d_1,\ldots,d_n)$, and the Hilbert series of $J'(d_1,\ldots,d_n)/\mathfrak mJ'(d_1,\ldots,d_n)$ is $F(d_1,\ldots,d_n)$. The function $F$ is the generalized $q,t$-Catalan number; the conjecture is proved for $n=3$ and remains open in general.