Curve neighborhoods and combinatorial property $\mathcal{O}$ for a family of odd symplectic partial flag manifolds

Let $E$ be an odd dimensional complex vector space and $IF:=IF(1,2;E)$ be the family of odd symplectic partial flag manifold. In this paper we give a full description of the irreducible components of the degree $d$ curve neighborhood of any Schubert variety of $IF$, study their lattice structure, and prove a combinatorial version of Conjecture $\mathcal{O}\mathrm{.}$