Prolate quadrature conjecture for exponential-function errors

Let $c>0$ be a real number and let $n>2c/\pi$ be an integer. Let $t_1,\dots,t_n$ and $W_1,\dots,W_n$ be the nodes and weights of the quadrature introduced in the source. Let $a\in[-1,1]$ be real, and let $\lambda_n$ be the associated eigenvalue.

Exponential quadrature-error conjecture. The quadrature error for the exponential function satisfies

$$\int_{-1}^1e^{icax}\,dx-\sum_{j=1}^ne^{icat_j}W_j=O\left(|\lambda_n|^2\sqrt{n}\right).$$

The estimate is motivated by the preceding analysis of the prolate functions and numerical observations, which suggest that the error is of order $|\lambda_n|$ in an intermediate regime. The supplied text does not report a proof or disproof of this sharper exponential-error estimate.