Consider the maximum $M_N of $\log |det(zI-U_N)$ for $z\in S^1$, where $U_N$ is a C$\beta$E random matrix. Fyodorv,Hiary and Keating conjectured (for $\beta=2$) that with $m_N=\log N-(3/4)\log\log N$, $\sqrt{2\beta}(M_N-\sqrt{2/\beta}m_N)$ converges to an explicit limit law, which coincides with the sum of two independent Gumbel(1) variables. Paquette and I proved that the limit is equal in law to the sum of a Gumbel(1) random variable and an independent random variable which follows the law of the logarithm of the total mass of a critical, a-priori non -Gaussian, multiplicative chaos.
I will describe work in progress with Lambert, Najnudel and Paquette which identifies that limit as a Gumbel(1) random variable, completing the proof of the FHK conjecture.