The secant-flag monotone conjecture

Let $\gamma$ be the rational normal curve, and for a set $S_i=\{s_{i1},\dotsc,s_{in}\}$ of $n$ distinct points of $\mathbb{R}\mathbb{P}^1$, let $F_\bullet(S_i)$ be the flag whose $r$-dimensional member is spanned by the first $r$ points of the corresponding set. Say that $(S_1,\dotsc,S_m)$ is monotone with respect to a Grassmannian Schubert problem $(w_1,\dotsc,w_m)$ if the sets lie in pairwise disjoint intervals whose associated points are monotone with respect to the problem. Secant-flag conjecture. For a Grassmannian Schubert problem on $\mathbb{F}\ell(\alpha;n)$, the intersection

$$X_{w_1}F_\bullet(S_1)\cap X_{w_2}F_\bullet(S_2)\cap\dotsb\cap X_{w_m}F_\bullet(S_m)$$

is transverse and all points of intersection are real whenever $(S_1,\dotsc,S_m)$ is monotone. Results of Eremenko, Gabrielov, Shapiro, and Vainshtein prove special cases; the general statement remains open.