The Bruhat trajectory conjecture for non-split real forms

Let $W(\mathfrak g,\mathfrak k)$ be the Weyl group associated with a non-split real form, and let $w,w'\in W(\mathfrak g,\mathfrak k)$ be invariant points of the Toda system. For a Bruhat comparison, write

$$w=\sigma_{\alpha_{i_k}}\cdot\dots\cdot\sigma_{\alpha_{i_1}}\cdot w'.$$

Bruhat trajectory conjecture. There exist trajectories connecting $w$ and $w'$ if and only if the corresponding elements are comparable in Bruhat order. In that case, the space of such trajectories coincides with the intersection of the Bruhat cell and the dual Bruhat cell, and its dimension is

$$\sum_{i=1}^k\dim\mathfrak g_{\alpha_i}.$$

In particular, this sum does not depend on the choice of the reflections $\sigma_{\alpha_i}$ in this formula.

This conjecture extends the trajectory-intersection description from split real forms to non-split real forms, where real Bruhat cells need not be identified directly with the real parts of complex cells and root spaces may have dimension greater than one.