The Ancestor Conjecture for finite Coxeter groups

Let $W$ be a Coxeter group with simple reflections $R$, and let $w\in W$. A prefix of $w$ is an element $u$ for which some reduced expression for $u$ is an initial segment of a reduced expression for $w$. Define the set of involution prefixes by

$$\mathcal{IP}(w)=\{u\in W:u\text{ is a prefix of }w\text{ and }u^2=1\},$$

and define the set of ancestors by

$$A(w)=\{u\in\mathcal{IP}(w):\ell(u)=\max_{v\in\mathcal{IP}(w)}\ell(v)\}.$$

A non-identity element $w$ has the ancestor property when $|A(w)|=1$, and a Coxeter group has the ancestor property when every non-identity element does.

The Ancestor Conjecture. Every finite Coxeter group has the ancestor property.

Equivalently, every interval $[1,w]$ in the weak left Bruhat order, for $w\ne 1$, contains a unique highest involution. The conjecture asserts the existence and uniqueness of a canonical maximal-length involution prefix for every non-identity element; its resolution is not supplied in the source.