The critical symplectic-group moment-of-moments conjecture

Let $P_{2N}(\theta)$ be the characteristic polynomial of a Haar-distributed symplectic matrix in $\mathrm{Sp}(2N)$. For $k>(1+\sqrt{5})^2/4$ and $s=1/\sqrt{k}$, define

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

Critical symplectic-group conjecture. For these parameters,

$$\operatorname{MoM}_{\mathrm{Sp}(2N)}(k,s)\sim\frac{k-1}{\Gamma(1-1/k)^k}\left(\frac{1}{2\pi}\int_0^{2\pi}|1-e^{2i\theta}|^{k-\sqrt{k}-2}\,d\theta\right)\left[\frac{G(1+1/\sqrt{k})^2}{G(1+2/\sqrt{k})}\right]^k2N\log N.$$

The threshold is imposed so that the predicted integral remains finite despite endpoint singularities in the symplectic density. The source offers this as a GMC-based heuristic, and the asymptotic remains unproved.