The Monotone Secant Conjecture for flag manifolds

Let $\gamma\colon\mathbb{R}\to\mathbb{R}^n$ be a rational normal curve, and let $\mathbb{F}\ell(\alpha;n)$ be a flag manifold. For a Schubert problem $(\sigma_1,\ldots,\sigma_m)$, let $X_{\sigma_i}F_{\bullet}^i$ denote the corresponding Schubert varieties, and call a list of secant flags disjoint when their intervals of secancy are pairwise disjoint and monotone when the order of those intervals agrees with the order of the types of the Grassmannian conditions.

Monotone Secant Conjecture. For any Grassmannian Schubert problem $(\sigma_1,\ldots, \sigma_m)$ on $\mathbb{F}\ell(\alpha;n)$ and any disjoint secant flags $F_{\bullet}^1,\ldots,F_{\bullet}^m$ that are monotone with respect to the Schubert problem, the intersection

$$X_{\sigma_1}F_{\bullet}^1\cap X_{\sigma_2}F_{\bullet}^2\cap\dotsb\cap X_{\sigma_m}F_{\bullet}^m$$

is transverse and all its points are real. This conjecture extends the Secant Conjecture from Grassmannians to partial flag manifolds; extensive computational experiments support it, but the general assertion remains open.