Conjugacy invariance under root-avoiding simple reflection shifts

Let $G$ be a twisted reductive group over an algebraically closed field, with Borel subgroup $B$. Let $W$ be its corresponding Weyl group, let $w\in W$, and let $s_i$ be a simple reflection such that $\alpha_i\notin V_w$. Set $w':=s_iws_i$. Root-avoiding shift conjecture. For every conjugacy class $C$ in $G$,

$$C\cap BwB\neq\varnothing\qquad\textrm{if and only if}\qquad C\cap Bw'B\neq\varnothing.$$

This asserts that the relevant Bruhat-double-coset incidence is unchanged under the specified conjugation step; no resolution evidence is supplied in the candidate text.