Generalized Dipper–James–Murphy conjecture for Kleshchev multipartitions

Let $\mathcal{P}_r(n)$ be the set of $r$-partitions of $n$, let $\mathcal{K}_r(n)$ be the set of Kleshchev $r$-partitions with respect to $(e;\mathbf{Q})$, and let an $r$-partition be $(\mathbf{Q},e)$-restricted if it admits a standard tableau whose residue sequence is distinct from the residue sequence of every standard tableau of every strictly smaller multipartition. Generalized Dipper–James–Murphy conjecture. For every $\boldsymbol\lambda\in\mathcal{P}_r(n)$, $\boldsymbol\lambda\in\mathcal{K}_r(n)$ if and only if $\boldsymbol\lambda$ is $(\mathbf{Q},e)$-restricted. The equivalence is known for $r=1$ and $r=2$, while the general case $r>2$ remains open.