Distinct-eigenvalue conjecture for subcritical covariance matrices

Let $\Sigma_n$ be the covariance matrix of the length power functionals in the subcritical regime, for $n\geq 2$, with eigenvalues $\lambda_1\leq\ldots\leq\lambda_n$. Distinct-eigenvalue conjecture. The eigenvalues are pairwise distinct:

$$\lambda_i\neq\lambda_j\quad\text{for all }i\neq j\in[n].$$

The conjecture is motivated by the question of whether the previously obtained lower and upper eigenvalue bounds are sharp for $n\geq3$; the supplied text gives no resolution of the distinctness claim.