The Lindelof Hypothesis may have been proved
In work published at the Arxiv (latest version March 2018, previous version November 2017) Professor Athanassios Fokas of Cambridge University states that he has proved it. In the first version of his paper (August 2017), he offers a proof of a "slight variant"; in the second and third versions he says it is possible to get from there to a proof of the Lindelof hypothesis itself. The publication of the required step is stated to be under preparation in a linked paper co-authored by himself and two other researchers.
Given the stature of the claimant, the claim is in a different category from the many that are made by mathematicians not previously known for having published weighty peer-reviewed results who write that they have proved famous hypotheses.