Irreducibility conjecture for codimension-one intersections of Springer fiber components

Let $\mathcal{B}_x$ be a Springer fiber of type $A$, and let $\mathcal{V}_T$ and $\mathcal{V}_S$ be two irreducible components of $\mathcal{B}_x$. Consider the case in which their intersection has codimension one in either component.

Irreducibility conjecture. The intersection in codimension one of two irreducible components of a Springer fiber of type $A$ is irreducible.

The conjecture extends the known irreducibility of intersections in the hook case to arbitrary nilpotent orbits. The paper proves this assertion for the two-column case, while the general type $A$ case remains open.