When is a Specht ideal Cohen-Macaulay?

For a partition $\lambda$ of $n$, let $I_{\lambda }^{\mathrm{S}\mathrm{p}}$ be the ideal of $R=K[x_1,\dots ,x_n]$ generated by all Specht polynomials of shape $\lambda$. We show that if $R\mathrm{/}{I}_{\lambda }^{\mathrm{S}\mathrm{p}}$ is Cohen–Macaulay then $\lambda$ is of the form either $(a,1,\dots ,1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R\mathrm{/}{I}_{(n-3,3)}^{\mathrm{S}\mathrm{p}}$ is NOT Cohen–Macaulay if and only if $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(K)=2$.