Subcritical Gaussian multiplicative chaos Fourier coefficient limit conjecture

Let $\gamma\in(1/\sqrt{2},\sqrt{2})$, let $\mu_\gamma$ be the Gaussian multiplicative chaos measure on $[0,1]$ with parameter $\gamma$, and let $c_n$ be its $n$-th Fourier coefficient. Subcritical Fourier coefficient limit conjecture. The random variables

$$(\log n)^{3\gamma/(2\sqrt{2})}n^{(\sqrt{2}-\gamma)^2/2}|c_n|$$

should converge in distribution, as $n\to\infty$, to a non-trivial limiting random variable. This is a heuristic prediction for the subcritical range $\gamma\in(1/\sqrt{2},\sqrt{2})$; the surrounding discussion explicitly says that the heuristics are not made rigorous there.