The maximal order of the zeta-function on the critical line

How large is the maximum value of $\vert\zeta(1/2+it)\vert$ for $T<t<2T$? It is known that the Riemann Hypothesis implies that the maximum is at most $\exp(c\log T/ \log\log T)$ for some $c>0$. It is also known that the maximum gets as big as $\exp(c_1(\log T/\log \log T)^{1/2})$ for a sequence of $T\to \infty$ for some $c_1>0$. It has been conjectured that the smaller bound (the one that is known to occur) is closer to the truth. However, the new conjectures about moments suggest that it may be the larger.

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