Lusztig's double-coset conjecture for minimally dominant elements

Let $W$ be the Weyl group of a maximal torus contained in a Borel subgroup $B$ of a twisted reductive group over an algebraically closed field. Let $w$ be a minimal-length element of $W$, let $w'$ be a minimally dominant element in the conjugacy class of $w$, and let $C$ be a conjugacy class of the reductive group. Lusztig's double-coset conjecture.

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

The claim proposes that replacing a minimal-length Weyl-group representative by a minimally dominant representative preserves which conjugacy classes meet the corresponding Bruhat double coset; the supplied text labels it among open problems.