Shapiro–Shapiro conjecture for real Schubert intersections

Let $m,p\geq 1$, let $\gamma$ be the real rational normal curve, and let $w_1,\ldots,w_n$ be Schubert conditions on $p$-planes in $(m+p)$-space. For points $s_1,\ldots,s_n\in\gamma$, write $\sigma_{w_i}(s_i)$ for the corresponding Schubert varieties. Shapiro–Shapiro conjecture. If the $s_i$ are real and

$$\sigma_{w_1}(s_1)\cap\sigma_{w_2}(s_2)\cap\cdots\cap\sigma_{w_n}(s_n)$$

is zero-dimensional, then all points of intersection are real. The theorem in the paper proves part of this conjecture, and the source reports substantial computational and theoretical evidence, but the general statement remains open.