What is known about the order of $\zeta(1+it)$?

Iwaniec-Kowalski gives

$|\zeta(1+it)| \lesssim (\log t)^{2/3},$

and oppositely Titchmarsh gives

$|\zeta(1+it)| \gtrsim \log \log t$

for infinitely many values of $t$.

Is this the limit of our knowledge? Is it conditionally known (or even expected) that

$|\zeta(1+it)| = e^{o(\log \log t)}$?