Full reality conjecture for Grassmannian Schubert data on partial flag manifolds

Let ${\bf a}=1<a_1<a_2<\cdots<a_r<a_{r+1}=n$, and let $(\alpha^1,i_1),\ldots,(\alpha^s,i_s)$ be Grassmannian Schubert data for the partial flag manifold $\mathbb{F}\ell_{\bf a}$, with $\alpha^j\in\binom{[n]}{i_j}$ and total codimension equal to $\dim\mathbb{F}\ell_{\bf a}$. For distinct real numbers $t_1<\cdots<t_s$, let the corresponding Schubert varieties be defined using flags osculating the rational normal curve. Grassmannian full-reality conjecture. If the indices are ordered monotonically, the intersection is transverse and all points are real for every such choice of parameters. If the indices are neither ordered nor cyclically ordered, there are choices giving a transverse intersection with all points real and choices giving a transverse intersection with not all points real. In particular, enumerative problems involving Grassmannian Schubert varieties on $\mathbb{F}\ell_{\bf a}$ are fully real. The source presents this as an emerging expectation based on computation and does not give a resolution.