Cfrgtkky

Let $W$ be a Weyl group (maybe better a Coxeter group, i.e. a group with action on vector space $V$ generated by reflections with some conditions). Consider $v in V$ be a vector. Let $text{Stab}_v subset W$ be a stabilizer of vector $v$.

Theorem
Group $text{Stab}_v$ is generated by reflections.

This Theorem is stated for example here, Theorem 1 (iii).
(in the case of Weyl group $text{Stab}_v$ is a Weyl group of Lie subgroup).

If we choose simple reflections in $W$ we will get a Bruhat order (I am interested in strong Bruhat order). Hence $text{Stab}_v$ is generated by reflections we can choose a set of simple reflections for $text{Stab}_v$; it gives as an interior Bruhat order on $text{Stab}_v$ (which a priori has nothing to do with the Bruhat order on whole $W$).

Is it true that for any choice of simple reflections in $W$ we can
find a set of simple reflections in $text{Stab}_v$ such that
restriction of Bruhat order from $W$ to $text{Stab}_v$ coincides with the
interior Bruhat order on $text{Stab}_v$?