Realizability conjecture for orthopositroids

For a positroid $\mathscr{M}$ of type $(k,n)$, and for subsets $I,J\subseteq [n]$ of size $k-1$, define

$$A_{IJ}^{\pm}(\mathscr{M})=\big\{\ell\in[n]:I\ell,J\ell\in\mathscr{M}\text{ and }(-1)^{\ell-1}\epsilon_{I\ell}\epsilon_{J\ell}=\pm1\big\}.$$

The positroid $\mathscr{M}$ is an orthopositroid if, for every $I,J\in\binom{[n]}{k-1}$,

$$A_{IJ}^{+}(\mathscr{M})=\emptyset\quad\Longleftrightarrow\quad A_{IJ}^{-}(\mathscr{M})=\emptyset.$$

Realizability conjecture for orthopositroids. Every orthopositroid $\mathscr{M}$ of type $(k,n)$ is realizable: there exists $X\in\operatorname{OGr}_{+}(k,n)$ such that $\mathscr{M}=\mathscr{M}_X$.

The motivation is that the positroid associated with any point of $\operatorname{OGr}_{+}(k,n)$ is necessarily an orthopositroid, because the Plücker coordinates satisfy the orthogonal relations. The paper states this realizability claim while noting that the general discussion of realizability is left for future work; no resolution is provided here.