The non-transverse monotone conjecture for Grassmannian Schubert problems

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$. Let $t_1,\dotsc,t_m\in\mathbb{R}\mathbb{P}^1$ be monotone with respect to the problem, meaning that the unique descents of the Grassmannian permutations vary monotonically along an orientation of $\mathbb{R}\mathbb{P}^1$. Non-transverse 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. The source explains that this conjecture implies the stronger reality conjecture, is supported by extensive computation, and remains open.