Cycles in subexpression graphs

Let $\mathfrak{S}(\underset{‾}{s},w)$ be the graph whose vertices are all subexpressions with target $w$ of a fixed expression $\underset{‾}{s}$ in generators of a Coxeter group and edges are the pairs of subexpressions with Hamming distance 2. We prove that $\mathfrak{S}(\underset{‾}{s},w)$ is connected and its cycle space is spanned by cycles of lengths $d+2$, where $d$ ranges over all positive divisors of all finite orders of products of at most two entries of $\underset{‾}{s}$.