The Gaussian value-distribution conjecture for the Hurwitz zeta function

Let $S\subset\mathbb{C}$ be Borel, and let $\zeta(s,\alpha)$ denote the Hurwitz zeta function for a shift parameter $0<\alpha<1$. Gaussian value-distribution conjecture. For large $T$ and almost all $\alpha$ with $0<\alpha<1$,

$$\frac{1}{T}\operatorname{meas}\left\{t\in[T,2T]:\frac{\zeta(\tfrac12+it,\alpha)}{\sqrt{\log T}}\in S\right\}\sim\frac{1}{2\pi}\iint_S e^{-(x^2+y^2)/2}\,dx\,dy.$$

This asserts that normalized critical-line values of the Hurwitz zeta function have a standard complex Gaussian distribution for almost every shift. It is motivated by the preceding higher-moment conjecture, and the source presents it as unresolved.