Bailey–Keating conjecture for moments of moments of the Riemann zeta function

Let $k,\gamma\in\mathbb{N}$ and $c>0$ be fixed, and define

$$\mathcal{M}_{k,2\gamma}(T)=\int_0^T\left(\frac{1}{2c}\int_{-c}^{c}|\zeta(1/2+it+ih)|^{2\gamma}\,dh\right)^kdt.$$

Let $a_{k,\gamma}$ be the arithmetic constant and $g_{k,\gamma}$ the geometric constant. Bailey–Keating conjecture.

$$\mathcal{M}_{k,2\gamma}(T)=a_{k,\gamma}g_{k,\gamma}T(\log T)^{k^2\gamma^2-k+1}\left(1+O_{k,\gamma}((\log T)^{-1})\right).$$

Moreover, when suitably smoothed, the quantity has the form

$$T P_{k^2\gamma^2-k+1}(\log T/2\pi)+O_{k,\gamma}(T^a),$$

where $P_{k^2\gamma^2-k+1}(x)$ has degree $k^2\gamma^2-k+1$ and $a<1$. This connects zeta moments of moments with Gaussian multiplicative chaos and random matrix theory; the asserted asymptotics remain conjectural.