The Lindelöf hypothesis is:
$$
\forall \epsilon >0,\exists C_\epsilon >0,\forall t\ge 1,\quad
\vert\zeta(\frac12+it)\vert\le C_\epsilon t^\epsilon.\qquad \tag{LH}.
$$
It is a weaker statement than the Riemann hypothesis: $(RH)\Longrightarrow (LH)$. The (not-so-easy) texbook result that the estimate above is true for $\epsilon=1/6$ was improved in 1986 by Bombieri and Iwaniec (mathreview#:MR0881101) who found the estimate for $\epsilon=\frac{9}{56}$.
Several works followed, using some variations of their method, but I do not think that the threshold $1/7$ was reached. 

Now my question:
is there a stronger  inequality, e.g. $$
\exists C >0,\forall t\ge 2,\quad\vert\zeta(\frac12+it)\vert\le C(\ln t)^C\tag{LH$^\sharp$}
$$
which would be equivalent to $(RH)$?