The invariant centralizer conjecture for parabolic Schur algebras

Let $D(n,r)^V$ denote the subspace of the parabolic Schur algebra consisting of elements fixed by the relevant $V$-action, and let $\bbC\Psi(\mathfrak{S}_r)$ be the complex span of the image of the symmetric group $\mathfrak{S}_r$ under the representation $\Psi$. Invariant centralizer conjecture. When $n\geq r$, $D(n,r)^V$ coincides with

$$\mathbb{C}\Psi(\mathfrak{S}_r).$$

The preceding argument verifies the assertion in the case $r=2$ and $n=1$ only in the surrounding discussion, while the conjecture proposes the stated equality for all $n\geq r$. Its status is not resolved in the supplied text.