The orbital variety closure conjecture for $\mathfrak{sl}_n$

Let ${\mathfrak g}={\mathfrak{sl}}_n$ and let ${\mathcal V}$ be an orbital variety, that is, an irreducible component of ${\mathcal O}\cap{\mathfrak n}$ for a nilpotent orbit ${\mathcal O}$ and a fixed nilradical ${\mathfrak n}$. Its closure $\overline{\mathcal V}$ is a closed subvariety of ${\mathfrak g}$. Orbital variety closure conjecture. The closure of an orbital variety is a union of orbital varieties. The conjecture would give a geometric description of orbital variety closures in type $A$, complementing the known descriptions of nilpotent orbit closures. The source presents this as being supported by special cases and computations in low rank, while the general assertion is left as a conjecture.