Dominance-order conjecture for permutahedron Laplacian eigenvalues

Let $n$ be a positive integer, let $\gamma \vdash n$ range over partitions of $n$, and let $\Lambda_\gamma$ be the set of Laplacian eigenvalues associated with the isotypic component indexed by $\gamma$. Define

$$\tilde{\lambda}_\gamma:= \min_{\lambda \in \Lambda_\gamma}\{\lambda\}.$$

Dominance-order conjecture. If $\nu \vartriangleright \gamma$, then

$$\tilde{\lambda}_\nu < \tilde{\lambda}_\gamma.$$

This conjectures that the smoothest eigenvectors in isotypic components earlier in the dominance ordering have strictly smaller Laplacian eigenvalues. The source provides no resolution evidence, so the conjecture remains open.