Conjecture on PSWF quadrature errors being bounded by the next eigenvalue

Let $c>0$, let $n>2c/\pi$ be an integer, and let $m$ be an integer with $0\leq m<n$. Let $\delta_n(\psi_m)$ be the quadrature error for the prolate spheroidal wave function $\psi_m$, and let $\lambda_n$ be the corresponding prolate spheroidal eigenvalue. PSWF quadrature-error bound conjecture. One has

$$\delta_n(\psi_m)=\left|\int_{-1}^1\psi_m(s)\,ds-\sum_{j=1}^n\psi_m(t_j)W_j\right|\leq |\lambda_n|.$$

The conjecture sharpens the preceding rigorous but apparently cautious error estimate and is supported by experiments in which the error was approximately $|\lambda_n|/2$.