Reducedness 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 satisfying

$$mp=|w_1|+\cdots+|w_n|.$$

For distinct real points $s_1,\ldots,s_n\in\gamma$, consider the corresponding Schubert intersection scheme. Reducedness conjecture. The intersection scheme is reduced. The source says this holds in every known case, but remains conjectural in general.