The Shapiro conjecture for the orthogonal Grassmannian

Let $OG(n)$ be the orthogonal Grassmannian, let $F_\bullet(t)$ be the isotropic flag osculating a specified rational normal curve, and let $X_{\boldsymbol{\kappa}}F_\bullet$ denote the Schubert variety indexed by a strict partition $\boldsymbol{\kappa}$. A Schubert problem $({\boldsymbol{\kappa}}_1,\dotsc,{\boldsymbol{\kappa}}_m)$ satisfies

$$\|{\boldsymbol{\kappa}}_1\|+\dotsb+\|{\boldsymbol{\kappa}}_m\|=\dim OG(n)=\binom{n+1}{2}.$$

The Shapiro conjecture for $OG(n)$. If $s_1,\dotsc,s_m$ are distinct real numbers, then

$$X_{{\boldsymbol{\kappa}}_1}F_\bullet(s_1)\cap\dotsb\cap X_{{\boldsymbol{\kappa}}_m}F_\bullet(s_m)$$

is transverse and all its points are real. This is an orthogonal-Grassmannian analogue of the Grassmannian Shapiro conjecture; unlike the Grassmannian case, the source presents it as an open conjecture.