Is the following statement true for finite reflection groups?

Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$ and let $z$ be in the orbit of $y$. If $\langle x,y\rangle = \langle x, z\rangle$, then there exists $g$ in the stabiliser of $x$ such that $z = gy$.

An important addition: $x$ and $y$ are in the same chamber.