The odd-order three-partition conjecture for cycle-generated Cayley graphs

Let $n\geq 7$ be odd and let $I$ satisfy

$$\{n-1,n\}\subseteq I\subseteq\{2,3,\ldots,n\}.$$

For a partition $\lambda$ of $n$, write $\lambda_\lambda^I$ for the eigenvalue associated with the corresponding irreducible representation of $S_n$ in the Cayley graph context of the source. Odd-order three-partition conjecture. One has

$$\lambda_{(3,1^{n-3})}^I<\max\left\{\lambda_{(1^n)}^I,\lambda_{(n-2,1^2)}^I,\lambda_{(2^2,1^{n-4})}^I\right\}.$$

The inequality would exclude the partition $(3,1^{n-3})$ from attaining the second largest eigenvalue and supports the claim that only the three listed partitions can achieve it in this case; the source gives no resolution status here.