Fyodorov-Hiary-Keating proposed very precise asymptotics for the maximum of the Riemann zeta function in almost all intervals along the critical axis. After reviewing the origins of this conjecture through the random matrix analogy, I will explain a proof up to tightness, building on an underlying branching structure. This work with Louis-Pierre Arguin and Maksym Radziwill relies on a multiscale analysis and twisted moments of zeta.