The projective-dimension conjecture for initial ideals of $L_2(n_1,1)$

Let $n_1$ be a positive integer, and let $R_{n_1,1}$ be the ring and $I_{L_2(n_1,1)}$ the ideal appearing in the paper, with $\mathrm{in}_{\prec}(I_{L_2(n_1,1)})$ denoting its initial ideal with respect to $\prec$. Projective-dimension conjecture.

$$\mathrm{pd}\left(R_{n_1,1}/\mathrm{in}_{\prec}(I_{L_2(n_1,1)})\right)=n_1.$$

The conjecture predicts an exact formula for the projective dimension of this initial-ideal quotient; the supplied text gives no evidence that it has been resolved.