Parabolic subgroup of Weyl group

Question

Let $W$ be the Weyl group of a semisimple algebraic group $G$. $I$ be the simple roots. $J\subset I$ generate a parabolic subgroup of $W$ denote by $W_J$. $w^J$ is the shortest representative of $w$ in $W/W_J$. Suppose we have $u^J\le v^J$, here $\le$ is the Bruhat order. Let $M=\{ x\in W_J| u^{J}x\le v^{J}\}$. My question is whether $M=\{id \}$ always hold?

Answer 1

No. In $A_2$ with $I=\{s_1,s_2\}$, take $J=\{s_1\}$ and $u^J=id$ and $v^J=s_1s_2$. Then $M=W_J$ because $s_1\le s_1s_2$.