Rains–Vazirani's bar-operator conjecture for quasiparabolic sets

Let $X$ be a quasiparabolic set, and suppose that the heights of the elements in each $W$-orbit are bounded below. Let

\mathcal M$ and

\mathcal N$ be the corresponding modules over the Hecke algebra.

Rains–Vazirani's bar-operator conjecture. If $X$ is bounded below, then each of the modules

$$\mathcal M \quad\text{and}\quad \mathcal N$$

has a bar operator.

Bar operators are needed to define quasiparabolic Kazhdan–Lusztig bases, generalizing Deodhar's parabolic bases. The claim is equivalent to Rains and Vazirani's Conjecture 8.4; no resolution is supplied here.