The bigraded Hilbert-series conjecture for alternating polynomials

Let $I_n$ be the ideal generated by alternating polynomials in $R=\mathbb{C}[x_1,y_1,\ldots,x_n,y_n]$, and let $q$ and $t$ record the two gradings. Write $s_1^n$ for the $n$-fold product of the Schur function $s_1$, let $s_{(1^n)}$ denote the Schur function associated with the partition $(1^n)$, and let $\nabla$ be the operator on symmetric functions used in the modified higher $(q,t)$-Catalan theory. Bigraded Hilbert-series conjecture. The bigraded Hilbert series of $I_n$ is

$$\frac{1}{(1-q)^n(1-t)^n}\left\langle \nabla(s_1^n),s_{(1^n)}\right\rangle.$$

This is presented as a special case of the preceding conjectural description of the syzygies of $I_n$; the surrounding conjecture is stated to be verified for $n\leq 6$, but the supplied text does not separately state the resolution status of this Hilbert-series specialization.