Conjecture on monotonicity and floating-point accuracy of PSWF quadrature errors

Let $c>1$ be real, let $n>2c/\pi$ be an integer, and let $S_n$ be the stated quadrature rule. For even integers $m$ with $0\leq m<n$, let $\delta_n(\psi_m)$ be the quadrature error. Quadrature-error monotonicity conjecture. The function $\delta_n(\psi_m)$ is monotonically increasing as a function of even $m$ in the range $0\leq m<n$. Moreover, in double-precision calculations, $\delta_n(\psi_m)$ is zero up to machine precision for every $0\leq m<2c/\pi$. The conjecture records numerical observations that had not been fully investigated; the source points to a theorem giving a related upper bound and to further numerical evidence.