Non-unisingularity conjecture for the conjugate Specht module

Let $n\geq 5$ be odd, and let $\mathcal{S}^{(n-2,2)'}=\mathcal{S}^{(2,2,1,\dots,1)}$ be the conjugate Specht module for the symmetric group $S_n$. A permutation whose cycle type has a transposition and an $(n-2)$-cycle has order $2n-4$.

Non-unisingularity conjecture. The module $\mathcal{S}^{(n-2,2)'}$ does not afford $1$ as an eigenvalue on the class of such permutations. Every other element of $S_n$ has $1$ as an eigenvalue; equivalently, $\mathcal{S}^{(n-2,2)'}$ is not unisingular for $S_n$.

The exceptional class consists of odd permutations when $n$ is odd, so it does not meet $A_n$. Consequently, the restriction $\mathcal{S}^{(n-2,2)'}\downarrow_{A_n}$ is expected to be unisingular. The conjecture is presented as an evidence-based claim, and no resolution is supplied in the source.