Unit-circle zero conjecture for Gaussian β-ensemble moment coefficients

Let the coefficients of the moments $m_{2l}$ be regarded as polynomials in the ensemble parameter $\kappa$, after extracting their stated powers of $N$ and $\kappa$. The coefficients have a palindromic or anti-palindromic numerator with respect to $N$. Unit-circle zero conjecture. In addition to this palindromic or anti-palindromic property, the numerator of each coefficient with respect to $N$ has simple zeros, all satisfying

$$|\kappa|=1,$$

so that the zeros occur in complex-conjugate pairs. This observation is supported by all cases accessible in the paper, but no general proof is given.