Coxeter-group $D_k$ conjecture for symmetric non-hook diagrams

Let $\lambda$ be a symmetric Young diagram that is not a hook, let $G_\lambda$ be the associated group, and let $\dim\lambda$ denote the dimension of the corresponding representation. Set $k=\dim\lambda/2$. Coxeter-group $D_k$ conjecture. If $\lambda$ is a symmetric diagram that is not a hook, then $G_\lambda$ is isomorphic to the Coxeter group $D_k$. The source notes that this conjecture is verified using SageMath for $\lambda=(3,2,1)$; its general validity remains open.