Conjecture on cycles in the expression graph of a Coxeter group

Let $(W,S)$ be a Coxeter system, let $w\in W$, and let an expression for $w$ be a tuple $(s_1,s_2,\ldots,s_k)$ of elements of $S$ such that $w=s_1s_2\cdots s_k$. Let $\mathcal{E}(w)$ be the directed graph whose vertices are all expressions for $w$, with directed edges given by braid moves. The graph $\mathcal{R}(w)$ is the corresponding graph on reduced expressions.

Expression-graph conjecture. For every $w\in W$, the theorem of Bergeron, Ceballos and Labbé concerning cycles in $\mathcal{R}(w)$ remains valid when $\mathcal{R}(w)$ is replaced by $\mathcal{E}(w)$.

This proposes extending the result from reduced to arbitrary, generally non-reduced, expressions. Unlike $\mathcal{R}(w)$, the graph $\mathcal{E}(w)$ is generally infinite and has several connected components, so the conjecture concerns cycles in a substantially larger graph.