Strong exchange conjecture for twisted involution conjugacy classes

Let $W$ be a Coxeter group, let $I^+$ be the set of twisted involutions in the relevant extension, and let

$$R=\{wsw^{-1}:(w,s)\in W\times S\}.$$

Let $\mathcal K\subset I^+$ be a $W$-conjugacy class satisfying the condition that, for every $(r,w)\in R\times\mathcal K$,

$$\ell(rwr)=\ell(w)\quad\Longrightarrow\quad rwr=w.$$

Strong exchange conjecture. Under this condition,

$$\ell(rwr)<\ell(w)\quad\Longrightarrow\quad rwr<w$$

for every $(r,w)\in R\times\mathcal K$.

This is proposed as a strong exchange condition analogous to Hultman's weak exchange condition. The source gives no resolution.