We prove a lower bound on the maximum of the Riemann zeta function in a typical short interval on the critical line. Together with the upper bound from the previous work of the authors, this implies tightness of $$ \max_{|h|\leq 1}|ζ(\tfrac 12+{\rm i} τ+{\rm i} h)|\cdot \frac{(\log\log T)^{3/4}}{\log T}, $$ for large $T$, where $τ$ is uniformly distributed on $[T,2T]$. The techniques are also applied to bound the right tail of the maximum, proving the distributional decay $\asymp y e^{-2y}$ for $y$ positive. This confirms the Fyodorov-Hiary-Keating conjecture, which states that the maximum of $ζ$ in short intervals lies in the universality class of logarithmically correlated fields.