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 $End (\underline{1})=k$ 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.**