Syzygy Hilbert-series conjecture for the alternating ideal

Let $R=\mathbb{C}[x_1,y_1,\ldots,x_n,y_n]$, let $\mathfrak{m}$ be its homogeneous maximal ideal, and let $I_n\subset R$ be the ideal generated by alternating polynomials. For each $1\leq i\leq n$, consider the bigraded Hilbert series of

$$\operatorname{Tor}_i(R/I_n,\mathbb{C})=\operatorname{Tor}_i(R/I_n,R/\mathfrak{m}).$$

Syzygy Hilbert-series conjecture. This Hilbert series equals

$$(-1)^{i-1}\sum_{\substack{\lambda\vdash n\operatorname{spin}(\lambda')=i-1}}\langle (s_1)^n,s_\lambda\rangle\,\langle\nabla(s_\lambda),s_{(1^n)}\rangle.$$

The conjecture gives a combinatorial and symmetric-function description of the syzygies of $I_n$; it is verified in the source for $n\leq6$, while the general case remains open.