The critical orthogonal-group moment-of-moments conjecture

Let $P_N(\theta)$ be the characteristic polynomial of a Haar-distributed orthogonal matrix in $\mathrm{O}(N)$. For $k>1$ and $s=1/\sqrt{k}$, define

$$\operatorname{MoM}_{\mathrm{O}(N)}(k,s):=\mathbb{E}_{\mathrm{O}(N)}\left[\left(\frac{1}{2\pi}\int_0^{2\pi}|P_N(\theta)|^{2s}\,d\theta\right)^k\right].$$

Critical orthogonal-group conjecture. For every $k>1$ with $s=1/\sqrt{k}$,

$$\operatorname{MoM}_{\mathrm{O}(N)}(k,s)\sim\frac{k-1}{\Gamma(1-1/k)^k}\left(\frac{1}{2\pi}\int_0^{2\pi}|1-e^{2i\theta}|^{k+\sqrt{k}-2}\,d\theta\right)\left[\frac{G(1+1/\sqrt{k})^2}{G(1+2/\sqrt{k})}\right]^kN\log N.$$

The prediction applies the GMC critical-moment heuristic to the non-stationary Gaussian field associated with orthogonal characteristic polynomials. Its validity is not established in the source, particularly near the exceptional points $\theta=0,\pi$.