Suppose $(W,S)$ is a Coxeter system and let $<$ denote the (strong) Bruhat Order of $W$; that is $u < b$, there exists some sequence of $t_1,\ldots,t_k \in S^W$ such that $v = ut_1\ldots t_k$ and $l(v) = l(u) + k$ (for the usual length function $l$ with respect to $S$).
Given $w \in W$ and $\{t_1,\ldots,t_k\} \subseteq S^W$ is a set of pairwise commuting elements, is the following true:
$$ w < wt_i \quad \forall i =1,\ldots,k \iff w < wt_1\ldots t_k?$$
I have thought about this for a while and assume the answer should be obvious but I can't see it directly.