Radziwiłł's large-deviation conjecture for Selberg's central limit theorem

Let $\tau$ be uniformly distributed on $[T,2T]$, and set

$$\sigma_T^2=\frac{1}{2}\log\log T.$$

For $k>0$, consider the regime $V\sim k\sqrt{2\log\log T}$. Radziwiłł's conjecture. In this regime,

$$\mathbf{P}\Big(\log |\zeta(1/2+{\rm i}\tau)|>\sigma_TV\Big)\sim C_k\int_V^\infty\frac{e^{-x^2/2}}{\sqrt{2\pi}}\,{\rm d}x.$$

This predicts that the constants from the moment conjecture correct the Gaussian tail in Selberg's central limit theorem at deviations of the order of the variance. The statement is supported by the moment conjecture and by the paper's numerical and theoretical evidence, but remains unproved in general.