Caputo's shuffle spectral gap conjecture

Let $n$ be a positive integer with $n\ge 3$. For each $A\subset\{1,\ldots,n\}$, let $S_{n,A}$ be the set of permutations in $S_n$ fixing every element outside $A$, and define

$$J_{n,A}:=\sum_{\pi\in S_{n,A}}\pi.$$

For nonnegative coefficients $\alpha_A$, set

$$w=\sum_{A\subset\{1,\ldots,n\}}\alpha_AJ_{n,A}.$$

Let $\operatorname{supp}w$ denote the support of $w$, and let $\psi_{S_n}(w)$ be the spectral gap of $w$, while $\psi_{S_n}(w,\mathsf D)$ is the spectral gap in the representation $\mathsf D$. Caputo's conjecture. If $\operatorname{supp}w$ generates $S_n$, then

$$\psi_{S_n}(w)=\psi_{S_n}(w,\mathsf D).$$

This conjecture was communicated privately by P. Caputo, and the source reports that it resisted both proof attempts and numerical disproof; no resolution is supplied.