Parity conjecture for Coxeter braids

For integers $0\leq d_1\leq d_2\leq\cdots\leq d_n$, define the Coxeter braid

$$\beta(d_1,\dots,d_n)=\operatorname{JM}_n^{d_1}\operatorname{JM}_{n-1}^{d_2}\cdots\operatorname{JM}_2^{d_{n-1}},$$

where $\operatorname{JM}_k$ is a Jucys–Murphy element on the first $k$ strands. A braid is parity when its triply graded Khovanov–Rozansky homology $\mathrm{HHH}$ is supported only in even homological degrees. Parity conjecture. The Coxeter braid $\beta(d_1,\dots,d_n)$ is parity for all $n$ and for all $0\leq d_1\leq\cdots\leq d_n$. The conjecture is known for Coxeter braids on three and four strands, while the assertion for arbitrary numbers of strands remains open.