Let $W$ be the Weyl group of a root system $\Phi$ with base $\Delta$ and system of positive roots $\Phi^+$. Let $S = \{ w_{\alpha} : \alpha \in \Delta \}$ be the set of simple reflections corresponding to elements of $\Delta$. Let $\theta \subset \Delta$, and let $w_0 = w_l w_{l,\theta}$, where $w_l$ and $w_{l,\theta}$ are the long elements of $W$ and $W_{\theta} = \langle w_{\alpha} : \alpha \in \theta \}$.

The element $w_0$ is characterized uniquely by the following property: in each left coset of $W_{\theta}$ in $W$, there is a unique representative of smallest length, and $w_0$ is the longest of all these special representatives. It has the property that $\ell(w_0x) = \ell(w_0) + \ell(x)$ for all $x \in W_{\theta}$.

I noticed it follows immediately from here that if $x \in W_{\theta}$, then $w_0x \geq w_0$ in the Bruhat order. I was wondering whether the converse is true. So my question is:

If $w \in W$, and $w \geq w_0$ in the Bruhat order, then do we have $w = w_0x$ for some $x \in W_{\theta}$?

Weyl groupis too restrictive; it just needs to be a finite Coxeter group. $\endgroup$