When is a Specht ideal Cohen-Macaulay?

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