The proximality conjecture for $\Theta$-positive elements

Let $G$ be a semisimple Lie group with a $\Theta$-positive structure, let $G/P_\Theta$ be its associated flag variety, and let $G_\Theta^{>0}$ denote the $\Theta$-positive semigroup. The proximality conjecture. Any $g\in G_\Theta^{>0}$ acts proximally on $G/P_\Theta$: it has a unique attracting and a unique repelling fixed point in $G/P_\Theta$, these points are transverse, the action on the tangent space at the attracting fixed point is strongly contracting, and the action on the tangent space at the repelling fixed point is strongly expanding. This is proposed as an analogue of properties of totally positive elements proved by Lusztig in split real Lie groups; the source does not state a resolution.