Radziwiłł's Gaussian value-distribution conjecture for log-modulus of the Riemann zeta-function

Let

$$\mathscr{S}(T,V)=\{t\in[T,2T]\mid \log|\zeta(1/2+it)|>V\}$$

be the set of ordinates for which the logarithm of the modulus of the Riemann zeta-function exceeds $V$, and let $\operatorname{meas}$ denote Lebesgue measure.

Radziwiłł's Gaussian value-distribution conjecture. If $V=o(\sqrt{\log\log T})$ as $T\to+\infty$, then

$$\frac{1}{T}\operatorname{meas}\left(\mathscr{S}\left(T,V\sqrt{\frac{1}{2}\log\log T}\right)\right) \sim \int_V^\infty e^{-u^2/2}\frac{du}{\sqrt{2\pi}}.$$

This is a central-limit-type prediction for the value distribution of $\log|\zeta(1/2+it)|$. The supplied text says that Radziwiłł proved the conjectural estimate in a shorter range, but does not establish the full stated range $V=o(\sqrt{\log\log T})$.