The maximum of the CUE field

Let $U_N$ denote a Haar Unitary matrix of dimension N, and consider the field $${\bf U}(z) = \log |\det(1-zU_N)|$$ for z in the unit disk. Then, $$\frac{\max_{|z|=1} {\bf U}(z) -\log N + \frac{3}{4} \log\log N} {\log\log N} \to 0$$ in probability. This provides a verification up to second order of a conjecture of Fyodorov, Hiary and Keating, improving on the recent first order verification of Arguin, Belius and Bourgade.