Caputo's hypergraph conjecture for symmetric-group representations

Let $\Gamma=([n],w)$ be a weighted hypergraph on $[n]$, with non-negative weights, and let $\rho$ be a non-trivial irreducible representation of $\mathrm{Sym}(n)$. Write $\pi_{(n-1,1)}$ for the standard representation minus the trivial representation, and let $\lambda_{\min}(\Gamma,\rho)$ denote the smallest eigenvalue associated with $\rho$. Caputo's hypergraph conjecture. For every such $\Gamma$ and $\rho$,

$$\lambda_{\min}(\Gamma,\pi_{(n-1,1)})\leq\lambda_{\min}(\Gamma,\rho).$$

This generalizes the known theorem for measures supported on transpositions. The conjecture is still open in general.