Generalised Foulkes conjecture

Let $a,b,c,d\in\mathbb{N}$ satisfy $ab=cd$ and $a=\min(a,b,c,d)$. Write $U\leqslant V$ when $U$ is isomorphic to a subrepresentation of $V$. Generalised Foulkes conjecture.

$$\operatorname{Sym}^b\operatorname{Sym}^a\mathbb{C}^N\leqslant \operatorname{Sym}^d\operatorname{Sym}^c\mathbb{C}^N.$$

This four-parameter generalisation is attributed in the source to Vessenes and is presented as a conjectural representation-theoretic strengthening of Foulkes's conjecture. The paper proves cases under additional divisibility conditions, while the unrestricted statement remains open in the supplied text.