Let $(W, S)$ be a Coxeter system. Let $J \subseteq S$ and recall that $W_J = \langle s: s \in J\rangle \subseteq W$.
Define $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$.
Proposition. For each $w \in W$ there is a unique $u \in W^J$ and $v \in W_J$ such that $w = uv$. Moreover, it holds for these elements that $\ell(w) = \ell(u) + \ell(v)$. Also, $u$ is the unique element of smallest length in the coset $wW_J = \{wx : x \in W_J\}$.
Question: Does $W^J\subseteq \langle t\in T: t\not\in W_J\rangle$, where $T$ is the set of reflections?