The characteristic bound conjecture for Cohen–Macaulay Specht ideals

Let $K$ be a field, let $R=K[x_1,\ldots,x_n]$, and let $\lambda=(\lambda_1,\ldots,\lambda_l)$ be a partition of $n$ satisfying condition (2) or (3) of the stated classification proposition, namely $l=2$, or $l=3$, $\lambda_1=\lambda_2$, and $\lambda_3=1$. Characteristic bound conjecture. The quotient $R/I^{\rm Sp}_\lambda$ is Cohen–Macaulay if and only if

$$\operatorname{char}(K)=0\quad\text{or}\quad \operatorname{char}(K)\ge n-\lambda_1.$$

The conjecture is motivated by computer experiments on the characteristic dependence of Specht ideals; the source does not report a proof or disproof.