Conjecture on the decay threshold for prolate spheroidal eigenvalues

Let $c>1$ and $0<\varepsilon<1$ be real numbers, and let $n>0$ be an integer. Let $\lambda_n$ denote the prolate spheroidal eigenvalue. Eigenvalue-decay conjecture. If

$$n>\frac{2c}{\pi}+10+\frac{2}{\pi^2}(\log c)\log\frac{1}{\varepsilon},$$

then

$$|\lambda_n|<\varepsilon.$$

This gives a simple empirical threshold ensuring eigenvalue decay and is presented alongside a more precise rigorous theorem; the conjecture concerns the usefulness of the simplified bound.