The orbit conjecture for real reduced double Bruhat cells

Let $G$ be a split semisimple algebraic group with Weyl group $W$, and let $L^{u,v}=N\overline{u}N\cap B_-vB_-$ be the reduced double Bruhat cell for $u,v\in W$. For a reduced word $\mathbf{i}\in R(u,v)$, write $\Gamma_{\mathbf{i}}(\mathbb{F}_2)$ for the associated group acting on $\mathbb{F}_2^{\ell(u)+\ell(v)}$. Orbit conjecture. For every two elements $u$ and $v$ in $W$, and every reduced word $\mathbf{i}\in R(u,v)$, the connected components of $L^{u,v}(\mathbb{R})$ are in a natural bijection with the $\Gamma_{\mathbf{i}}(\mathbb{F}_2)$-orbits in $\mathbb{F}_2^{\ell(u)+\ell(v)}$. In the case $G=SL_n$ and $v=w_0$, this was proved for a special reduced word, so the general assertion is a theorem in the stated special case but is not established here in full generality.