Gorsky–Rasmussen conjecture for twisted projectors

Let $n$ be a positive integer, let $w\in S_n$ be a permutation, and let $E$ be the ring from Definition of the source. Let $x_1,\ldots,x_n$ be formal indeterminates, and let $w(P_{1^{n}}^\vee)$ denote the complex obtained by applying $w$ to the dual one-column projector. Define $J_w$ to be the ideal generated by the differences $x_{w(i)}-x_i$ for $1\leq i\leq n$. Gorsky–Rasmussen conjecture. For every permutation $w\in S_n$, the degree-zero homology satisfies

$$\operatorname{HHH}^0(w(P_{1^{n}}^\vee))\cong E/J_w.$$

The claim gives an explicit flag-Hilbert-scheme description of the degree-zero homology of twisted projectors. The supplied text does not state whether this conjecture has been resolved.