Bruhat covering property for twisted involutions in Coxeter systems

Let $(W,S)$ be a Coxeter system with length function $\ell:W\to\mathbb{N}$ and Demazure product $\circ:W\times W\to W$. Suppose $w\mapsto w^*$ is an automorphism of $W$ with $S^*=S$. Define the set of twisted involutions by

$$I_* = \{w\in W:w^{-1}=w^*\}.$$

For $y\in I_*$, let $\mathcal{A}_*(y)$ be the set of elements of minimal length such that $(w^*)^{-1}\circ w=y$, and set

$$T=\{wsw^{-1}:w\in W,\ s\in S\}.$$

Bruhat covering property. If $y\in I_*$ is a twisted involution in an arbitrary Coxeter group and $t\in T$, then there exists at most one $z\in I_*$ such that

$$\{wt:w\in\mathcal{A}_*(y)\text{ and }\ell(wt)=\ell(w)+1\}\cap\mathcal{A}_*(z)\neq\varnothing.$$

The result extends the Bruhat covering property proved in the paper for involutions in affine symmetric groups and conjecturally applies to arbitrary Coxeter systems; its general validity remains open.