Conjecture on osculating Schubert families for classical flag manifolds

Let $V$ be a vector space with a bilinear form, and let $G:=\operatorname{Aut}(V,\langle\cdot,\cdot\rangle)$ be respectively a general linear, orthogonal, or symplectic group according as the form is identically zero, nondegenerate symmetric, or nondegenerate alternating. Let $\gamma$ be a real rational normal curve in $V$ whose osculating flags $F_\bullet(s)$ are isotropic in the orthogonal and symplectic cases. Let $P$ be a parabolic subgroup of $G$, let $W$ be the Weyl group, and let $W_P$ be the associated parabolic subgroup. For $w\in W/W_P$, let ${\mathcal X}_w\to\gamma$ be the family of Schubert varieties $X_wF_\bullet(s)$ in $G/P$. Osculating Schubert-family conjecture. For any $w\in W/W_P$, the family ${\mathcal X}_w\to\gamma$ respects the Bruhat decomposition of $G/P$ given by the flag $F_\bullet(0)$, and any collection of these families is in general position. The conjecture would extend the corresponding Grassmannian result of Eisenbud and Harris to flag varieties of the classical groups; the source notes that the relevant structure constants have few known formulas and that finding a combinatorial formula for them is open.