QP1–QP2 conjecture for conjugacy classes of twisted involutions

Let $(W,S)$ be a Coxeter system, let $I^+$ denote the set of twisted involutions in the relevant extension, and give it the height function

$$\mathrm{ht}=\frac{1}{2}\ell.$$

Consider a conjugacy class in $I^+$ satisfying property (QP1) of the source's definition of quasiparabolicity.

QP1–QP2 conjecture. Every conjugacy class in $I^+$ satisfying (QP1) also satisfies (QP2), and hence is quasiparabolic.

The claim is motivated by technical machinery for twisted involutions and is presented as an open conjecture in the source.