Let $W$ be a Coxeter group, and $S$ be its set of simple reflections. For any $w \in W$, define $\mathcal L(w)$
$\mathcal L(w) = \{s \in S| sw<w \}.$
Is it true:
If $w \in W$, $s_1,s_2 \in S$, the order of $s_1s_2$ is $\alpha$ (an integer $\geq 2$ or $\infty$), and $\mathcal L(w) =\{s_1,s_2\}$, then $\alpha$ is finite and there exists $w' \in W$ such that $w = (s_1s_2)^{\alpha}w'$, $\ell(w)=\ell(w')+\alpha$.