Lexicographic eigenvalue-ordering conjecture for the derangement graph

Let $n$ be a positive integer, let $\lambda\vdash n$ be a partition with first part $\lambda_1$, and let $\eta_\lambda$ denote the eigenvalue of the derangement graph $\Gamma_n$ indexed by $\lambda$. Let $\lambda^*\vdash n$ be the largest partition in lexicographic order among all partitions whose first part is $\lambda_1$.

Lexicographic eigenvalue-ordering conjecture. For every partition $\lambda=(\lambda_1,\ldots,\lambda_s)\vdash n$,

$$|\eta_{(\lambda_1,1^{n-\lambda_1})}|\leq |\eta_\lambda|\leq |\eta_{\lambda^*}|.$$

Thus, among partitions with a fixed first part, the absolute eigenvalues are conjectured to be bounded below by the hook partition and above by the lexicographically largest partition.

The conjecture is motivated by the preceding theorems and computations for small values of $n$. The supplied source gives no resolution, so its status remains open.