Drury's partition-support conjecture for Schur power eigenvalues

Let $\Pi$ be a partition of $n$, and let its Ferrers diagram be the associated diagram. For $A\in{\cal H}_n$, let the block of the Schur power matrix $\pi(A)$ associated with $\Pi$ be the block indexed by the corresponding irreducible representation of $\mathcal{S}_n$. Drury's partition-support conjecture. If the Ferrers diagram of $\Pi$ contains neither the Ferrers diagram of $(3,2)$ nor that of $(7,1)$, then the largest eigenvalue of the block associated with $\Pi$ is at most $\mathop{\rm per} A$. The source presents this as an open conjecture.