A note on involution prefixes in Coxeter groups

Let $(W,R)$ be a Coxeter system and let $w\in W$. We say that $u$ is a prefix of $w$ if there is a reduced expression for $u$ that can be extended to one for $w$. That is, $w=uv$ for some $v$ in $W$ such that $\mathrm{\ell }(w)=\mathrm{\ell }(u)+\mathrm{\ell }(v)$. We say that $w$ has the ancestor property if the set of prefixes of $w$ contains a unique involution of maximal length. In this paper we show that all Coxeter elements of finitely generated Coxeter groups have the ancestor property, and hence a canonical expression as a product of involutions. We conjecture that the property in fact holds for all non-identity elements of finite Coxeter groups.