I have been reading Deligne-Milne's Tannakian Categories and got to the point at the end of part 3 where they discuss what went wrong with Saavedra's definition. Motivated by their counterexample, they ask the following question:
Let $C$ be a $k$-linear rigid abelian tensor category over a field $k$. Let us further assume that $\hom (1,1)=k$ and there exists a fiber functor with values in a field extension $k^'\supseteq k$. Does it follow that $C$ is Tannakian?
What is the current status of this question? A reference to a solution or to a modern discussion would be very useful.