Divided-difference conjecture for cohomology classes of balanced two-row Springer fibers

Let $\mathcal{B}_{\boldsymbol{\rho}}$ be the component of the two-row Springer fiber indexed by a standard tableau $\boldsymbol{\rho}$ of shape $(\rho,\rho)$, with cup diagram $\mathcal{C}(\boldsymbol{\rho})$. Order its cups as $(i_1<j_1),\ldots,(i_\rho<j_\rho)$ so that every cup nested inside a cup appears earlier. Let $\mathfrak{S}_{w_0}(\mathbf{x})$ be the Schubert polynomial of the longest permutation, and let $\partial_i$ be the divided-difference operator. Divided-difference conjecture.

$$[\mathcal{B}_{\boldsymbol{\rho}}] = \left( \partial_{i_1} + \cdots + \partial_{j_1-1} \right) \left( \partial_{i_2} + \cdots + \partial_{j_2-1} \right) \cdots \left( \partial_{i_\rho} + \cdots + \partial_{j_\rho-1} \right)\mathfrak{S}_{w_0}(\mathbf{x}).$$

The paper proves the conjecture for a specific family and reports SageMath and Macaulay2 checks through $n=8$; the general formula remains open.