Sottile's general secant conjecture

Let $\nu\vdash n$ and let $\boldsymbol{\mu}\Vdash\kappa$, where $\kappa=(\kappa_1,\dots,\kappa_s)$ is a composition of $n$. Let $I_1,\dots,I_s\subseteq\mathbb{R}$ be pairwise disjoint real intervals. A generalized secant flag to the moment curve $\gamma$ along an interval is a complete flag whose subspaces are generalized secants to $\gamma$ along that interval. Sottile's general secant conjecture. If $F^{(1)}_\bullet,\dots,F^{(s)}_\bullet$ are generalized secant flags along $I_1,\dots,I_s$, respectively, then the corresponding Schubert intersection is real and scheme-theoretically reduced. The divisor case is proved in the paper, but the general form remains open.