The Monotone Osculating Conjecture for flag manifolds

Let $\gamma\colon\mathbb{R}\to\mathbb{R}^n$ be a rational normal curve, and let $F_{\bullet}(t)$ denote its osculating flag at $t$, whose $j$-dimensional subspace is spanned by $\gamma(t),\gamma'(t),\ldots,\gamma^{(j-1)}(t)$. For a Schubert problem $(\sigma_1,\ldots,\sigma_m)$ on $\mathbb{F}\ell(\alpha;n)$, let $X_{\sigma_i}F_{\bullet}^i$ be the associated Schubert varieties.

Monotone Osculating Conjecture. For any such Schubert problem and any flags $F_{\bullet}^1,\ldots,F_{\bullet}^m$ osculating $\gamma$ at points that are monotone with respect to the Schubert problem, the intersection

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

is transverse and all its points are real. The conjecture is presented as the osculating limit of the Monotone Secant Conjecture; special cases and substantial computational evidence were known, but the general assertion remained open in the source.