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 $(RH)$?