Ideal description conjecture for Coxeter braid homology

Let $0\leq d_1\leq\cdots\leq d_1$, let $\beta=\operatorname{JM}_n^{d_1}\ldots\operatorname{JM}_2^{d_{n-1}}$, and set $J=J(d_1,\ldots,d_n)$, where

$$J(d_1,\ldots,d_n)=\bigcap_{i<j}(t_i-t_j,x_i-x_j)^{d_i}\subseteq\mathbb C[x_1,\ldots,x_n,y_1,\ldots,y_n].$$

The notation $\mathrm{HHH}^{a=0}$ and $\mathrm{HY}^{a=0}$ denotes the $a=0$ pieces of triply graded Khovanov–Rozansky homology and its $y$-ified version, respectively. Ideal description conjecture. Then $\mathrm{HHH}^{a=0}(\beta)$ is parity and

$$\mathrm{HY}^{a=0}(\beta)\simeq J$$

and

$$\mathrm{HHH}^{a=0}(\beta)\simeq J/(y)J.$$

This is motivated by the Gorsky–Hogancamp description conditional on parity; the conjecture asserts parity and the resulting ideal identifications in the stated generality.