Weak Foulkes conjecture for multiset partitions

Let $a$ and $b$ be positive integers with $a<b$, and let $M$ be a multiset of size $ab$. A multiset partition divides $M$ into an unordered collection of submultisets. Weak Foulkes conjecture. The number of multiset partitions of $M$ into $b$ parts of size $a$ is at least the number of multiset partitions of $M$ into $a$ parts of size $b$. Equivalently, in the symmetric-function formulation given by the source, the relevant plethysm difference has nonnegative coefficients in the monomial symmetric-function basis, whereas full Foulkes conjecture asserts the analogous Schur-basis nonnegativity. This is a weaker combinatorial form of Foulkes's conjecture for tensor products of symmetric powers; its general status is not resolved in the supplied text.