Conjecture on prolate spheroidal quadrature error for exponential functions

Let $c>0$ and $a\geq 0$ be real numbers, and let $n>2c/\pi$ be an integer. Let $\delta_n(e^{icax})$ denote the quadrature error associated with the quadrature nodes $t_j$ and weights $W_j$, and let $\lambda_n$ be the prolate spheroidal eigenvalue. Exponential quadrature-error conjecture. If $0\leq a\leq 1$, then

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

If $1<a\leq 2$, then

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

This conjecture summarizes numerical observations about the size of the quadrature error in the two ranges of the frequency parameter $a$.