Coxeter-group $D_k$ conjecture for selected Young diagrams

Let $G_\lambda$ be the group associated with a Young diagram $\lambda$, and let $\dim\lambda$ denote the dimension of the corresponding representation. For $\lambda\in\{(3,2,1),(4,2,1^2),(3^2,2),(3^3)\}$, the computed order of $G_\lambda$ is $2^{k-1}k!$, where $k=\dim\lambda/2$. Coxeter-group $D_k$ conjecture. In these cases, $G_\lambda$ is isomorphic to the Coxeter group $D_k$. The claim is supported computationally for diagrams with at most nine cells and is verified using SageMath for $\lambda=(3,2,1)$, but remains a conjecture in general.