Critical multiplicative-chaos conjecture for zeta-function densities

Let $\beta_c=2$ and consider the random densities from the preceding conjecture, with the critical exponent $\beta=\beta_c$. Critical multiplicative-chaos conjecture. The preceding convergence conjecture should remain valid for $\beta=\beta_c=2$ after multiplying by the normalizing factor

$$(\log\log T)^{1/2}.$$

This is the critical analogue of the subcritical multiplicative-chaos prediction. The source gives no resolution status.