Fixed-point conjecture for the twist on the longest Richardson variety

Let $G=\operatorname{SL}_n(\mathbb{C})$, let $v=\operatorname{id}$ and $w=w_0$, and let $\twist_v^w$ be the twist automorphism of the open Richardson variety, preserving its totally positive part $\Rtp_v^w$. Fixed-point conjecture. The unique fixed point of $\twist_v^w$ inside $\Rtp_v^w$ is $gB_-$, where $g=(g_{i,j})_{i,j=1}^n$ is the upper triangular matrix with

$$g_{i,j}=\sqrt{\binom{n-i}{j-i}\binom{j-1}{j-i}},\qquad 1\leq i\leq j\leq n.$$

This is presented as an example in which the answer to the broader fixed-point question appears to be yes; the supplied text does not provide a proof or resolution.