Maximum of the Riemann zeta function on a short interval of the critical line

We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as $T \rightarrow \infty$ for a set of $t \in [T, 2T]$ of measure $(1 - o(1)) T$, we have $$ \max_{|t-u|\leq 1}\log\left|\zeta\left(\tfrac{1}{2}+i u\right)\right|=(1 + o(1))\log\log T . $$