Foulkes's conjecture for symmetrized weight spaces

Let $a$ and $b$ be positive integers with $a<b$. For a polynomial representation $W$ of $GL_n$ of degree $ab$, consider the symmetrized weight spaces corresponding to the weights

$$(a,a,\dots,a,0,0,\dots,0)$$

and

$$(b,b,\dots,b,0,0,\dots,0).$$

Here $S_n$ acts by permuting coordinates, and $S_n$-invariants mean the invariant subspaces for this action. Foulkes's conjecture. The space of $S_n$-invariants in the first symmetrized weight space has dimension at least that of the second, for every polynomial representation of $GL_n$ of degree $ab$. This is the mixed $GL_n$–$S_n$ formulation of Foulkes's conjecture, related to comparing modules induced from normalizers of Young subgroups and to the Schur-positivity of the corresponding plethysm difference. The conjecture remains open in general.