Totally positive secant conjecture for Schubert problems

Let $0\le k\le n$, and let $I_1,\dots,I_{k(n-k)}$ be pairwise disjoint intervals of $\mathbb{P}^1(\mathbb{R})$. For each $l$, let $X_l$ be a multiset of $k$ points in $I_l$, and let $W_l\in\operatorname{Gr}_{k,n}(\mathbb{R})$ be spanned by $\gamma(x),\gamma'(x),\dots,\gamma^{(p-1)}(x)$ for every $x\in X_l$, where $p$ is the multiplicity of $x$ in $X_l$. A real Grassmannian point is totally nonnegative, respectively totally positive, when all its Plücker coordinates are nonnegative, respectively positive, up to rescaling. Totally positive secant conjecture. If every $I_l\subseteq[0,\infty]$, the Schubert problem has $d_{k,n}$ distinct solutions $U\in\operatorname{Gr}_{n-k,n}(\mathbb{C})$, all real and totally nonnegative; if every $I_l\subseteq(0,\infty)$, it has $d_{k,n}$ distinct solutions, all real and totally positive. This is a totally positive analogue of the secant conjecture and implies it; the two parts are equivalent by a limiting argument, but the conjecture remains open.