Foulkes's conjecture for symmetric powers

Let $N$ be a positive integer, and let $m,n$ be integers with $m\leqslant n$. For representations of $\operatorname{SL}_N(\mathbb{C})$, write $U\leqslant V$ when $U$ is isomorphic to a subrepresentation of $V$. Foulkes's conjecture.

$$\operatorname{Sym}^m \operatorname{Sym}^n \mathbb{C}^N\leqslant \operatorname{Sym}^n \operatorname{Sym}^m \mathbb{C}^N.$$

This is a central special case of the plethysm problem and is stated in the source as open. It concerns comparing the decompositions of two iterated symmetric powers as $\operatorname{SL}_N(\mathbb{C})$-representations.