Primality conjecture for ideals defining two-row Springer fiber components

Let $n$ be a positive integer, let $\boldsymbol{\rho}$ be a standard Young tableau of shape $(n-\rho,\rho)$, and let $\mathcal{I}_{\boldsymbol{\rho}}$ be the associated ideal. Primality conjecture. For every positive integer $\rho$ with $1\leq \rho\leq \left\lfloor\frac{n}{2}\right\rfloor$, the ideal $\mathcal{I}_{\boldsymbol{\rho}}$ is prime. Computations in Macaulay2 support this assertion for two-row tableaux, but no proof or resolution is supplied here.