Typical short-interval bounds for the maximum of the Riemann zeta function

Let $\zeta(s)$ denote the Riemann zeta function. For $T>0$, consider its maximum on the interval $[T,T+2\pi]$.

Typical short-interval maximum conjecture. Let $\epsilon>0$. For $T_1$ sufficiently large depending on $\epsilon$, for all $T_1\leq T\leq 2T_1$, outside a set of “bad” values of $T$ of measure at most $\epsilon T_1$, one has

$$\exp(\log\log T-(2+o(1))\log\log\log T) \leq \max_{T\leq t\leq T+2\pi}|\zeta(1/2+it)| \leq \exp(\log\log T-(1/4+o(1))\log\log\log T).$$

This is motivated by comparing the prime-sum model for $\log|\zeta(1/2+it)|$ with a log-correlated random model on a typical short interval. The assertion gives separate lower and upper bounds rather than the sharper conjectural asymptotic above, and its status is not resolved here.