Lindelöf hypothesis claim I was randomly browsing, when I found this puff piece claiming a proof of the Lindelöf hypothesis by Fokas. Note that the Wikipedia article says that he claimed, then withdrew his claim in 2017, but the USC piece is dated June 25 2018. So, what is the truth?
 A: (Not an answer of any sort, just too long for a comment.)  The main result 
seems to be an integral equation (1.3) of the form 
$$\int_{-\infty}^\infty K(t,\tau) |\zeta(\tfrac{1}{2}+it\tau)|^2\,d\tau={\mathcal G}(t)$$
with  some explicit functions $K$ and ${\mathcal G}$.  This equation (if true) is presumably 
new, and may be interesting. 
However, in my view, how interesting it is would   depend  quite a bit on whether $|\zeta(\tfrac{1}{2}+it\tau)|^2$ is the  only  solution of it.  This sort of integral operators  may have kernels, and  if it is the  case then it would be rather  difficult to squeeze   the Lindelöf hypothesis  out of it.  (If I were the author then this is where I would look.)
P.S. For those who have  read ``puff piece'':  As pointed out by  Robert Israel, indeed,
in the (this far, latest) version 4 a proof of the Lindelöf hypothesis   is not claimed.
P.P.S Take the above with a pinch of salt; the last time I was involved with this subject was decades ago. 
A: I'm putting this as an answer because I haven't got enough reputation to post a comment.
In a paper published on 25 September 2018, written jointly with A Ashton, A Fokas asserts that a relation derived in that paper "provides the starting point of a novel approach which in a series of companion papers yields a formal proof of the Lindelöf hypothesis" (my emphasis).
A: From the announcement of the Séminaire Bourbaki du vendredi on 28/1/2022 :

