Enumerating cores of charged multipartitions

Granville and Ono proved that there is an $e$-core partition of $n$ for every $n\in \mathbb{N}$ if and only if $e\ge 4$, which translates to a statement about the existence of defect $0$ blocks for symmetric groups in positive characteristic and defect $0$ unipotent blocks of finite general linear groups in positive, non-defining characteristic. Motivated by analogous applications to the block theory of finite classical groups, imprimitive spetses, and cyclotomic Hecke algebras, we prove similar positivity statements about different variants of $e$-cores for charged multipartitions.