  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 $$ \max_{|h|\leq
1}|\zeta(\tfrac 12+{\rm i} \tau+{\rm i} h)|\cdot \frac{(\log\log T)^{3/4}}{\log
T}, $$ for large $T$, where $\tau$ 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 $\zeta$ in
short intervals lies in the universality class of logarithmically correlated
fields.
