The radicality conjecture for Specht ideals

Let $K$ be a field, let $R=K[x_1,\ldots,x_n]$, and let $I^{\rm Sp}_\lambda$ denote the Specht ideal associated to a partition $\lambda$ of $n$. Radicality conjecture. The Specht ideal $I^{\rm Sp}_\lambda$ is always a radical ideal. This would settle the radicality question for Specht ideals, which is relevant to determining their Cohen–Macaulayness via results for radical ideals; the source provides no resolution.