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

Iwaniec-Kowalski gives (pp. 226 citing a result of Vinogradov-Korobov)

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

and oppositely Titchmarsh gives (pp. 188 citing Bohr-Landau)

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

for infinitely many values of $t$.

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

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

[As David points out below, on RH the result in Titchmarsh's book is optimal.]