The iterated-projective-line-bundle conjecture for intersections of two-row Springer-fiber components

Let Springer fibers be associated with two-row Young shapes, and let their irreducible components be described by the corresponding tableau and cup-diagram data. An iterated $\mathbb{CP}^1$-bundle is a variety obtained through successive fibrations whose fibers are $\mathbb{CP}^1$. Two-row intersection conjecture. Pairwise intersections of components of Springer fibers of two-row type are iterated $\mathbb{CP}^1$ bundles. It would suffice to show that each pairwise intersection admits a description of the same form as the characterization of an individual component: the intersection consists of the flags satisfying the descriptions of both components simultaneously, and this admits a single description of that form. This conjecture is motivated by the preceding calculations; its status is not addressed beyond the supplied evidence, while the paper explains the proposed geometric description that would prove it.