Unimodality conjecture for the Hilbert series of $R_n({\mathbf a})$

Let $r\leq n$ and ${\mathbf a}\in({\mathbb Z}_{\geq0})^r$. Let $R_n({\mathbf a})$ be the doubly graded ring, with $R_n({\mathbf a})_{i,j}$ its component of $x$-degree $i$ and $\theta$-degree $j$, and define

$${\mathrm {Hilb}}(R_n({\mathbf a});q,z)=\sum_{i,j}\dim R_n({\mathbf a})_{i,j}q^iz^j.$$

Unimodality conjecture. The matrix of coefficients of ${\mathrm {Hilb}}(R_n({\mathbf a});q,z)$ has unimodal rows and columns. This would extend the observed symmetry and unimodality patterns of the doubly graded Hilbert series, but the source supplies no resolution.