Generalized Foulkes-type plethysm inequality

Let $m,n$ be positive integers, let $u$ be a partition of $n$, and let $u'$ denote its conjugate partition. For a partition $\tau$ of $mn$, write $p^{\tau}_{\nu,(m)}$ for the corresponding plethysm coefficient. Generalized Foulkes-type conjecture. If $\lambda\vdash mn$, then

$$p^\lambda_{\nu,(m)}\leq p^{\lambda+(n)}_{\nu,(m+1)}.$$

This is presented as a generalization of Foulkes' Second Conjecture. It is proved in the case $\nu=(1^n)$, but remains conjectural in general.