Uniform representation stability for cohomology of pure string motion groups

Let $P\Sigma_n$ be the pure string motion group and let $W_n$ be the hyperoctahedral group acting on it by permuting labels and reversing orientations. For each fixed $i\geq 0$, consider the rational cohomology representations $H^i(P\Sigma_n;\mathbb{Q})$. Uniform representation stability conjecture. For each fixed $i\geq 0$, the sequence $\{H^i(P\Sigma_n;\mathbb{Q})\}$ of $W_n$-representations is uniformly representation stable. This extends the representation-stability phenomena established earlier in the paper for braid-type groups to pure string motion groups; the source gives no resolution of the conjecture.