The monotone conjecture for arbitrary Schubert problems on flag manifolds

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $(w_1,\dotsc,w_m)$ be a Schubert problem for $\mathbb{F}\ell(\alpha;n)$. For each permutation $w_i$, let $\delta(w_i)$ be its set of descents. A list $(t_1,\dotsc,t_m)\in(\mathbb{R}\mathbb{P}^1)^m$ is monotone with respect to $(w_1,\dotsc,w_m)$ when the map $t_i\mapsto\delta(w_i)$ is monotone for an ordering compatible with some orientation of $\mathbb{R}\mathbb{P}^1$. General monotone conjecture. The intersection

$$X_{w_1}(t_1)\cap X_{w_2}(t_2)\cap\dotsb\cap X_{w_m}(t_m)$$

is transverse and all its points of intersection are real whenever the points are monotone with respect to $(w_1,\dotsc,w_m)$. The conjecture extends the Grassmannian formulation to arbitrary Schubert problems; the source notes examples with no monotone points and reports computational evidence, but the general assertion remains open.