The smoothing-scale conjecture for zeta maxima

Let $T>1$, let $\tau$ be chosen uniformly from $[T,2T]$, and let $0<\alpha<1$ with $\theta\sim(\log\log T)^{-\alpha}$. Consider an interval of size

$$(\log T)^\theta=\exp\bigl((\log\log T)^{1-\alpha}\bigr).$$

The smoothing-scale conjecture. For some tight sequence of random variables $(\mathcal M_\alpha(T),T>1)$,

$$\max_{|h|\leq\exp((\log\log T)^{1-\alpha})}\left|\zeta\left(\frac12+\mathrm{i}(\tau+h)\right)\right|=\frac{(\log T)^{\sqrt{1+\theta}}}{(\log\log T)^{\frac{1+2\alpha}{4}}}e^{\mathcal M_\alpha(T)}.$$

The conjecture proposes the scale at which the subleading-order transition between fixed positive $\theta$ and $\theta=0$ becomes smooth. It is motivated by the random model studied in the paper and remains open.