This intriguing question is due to Sorin Popa, so I'm marking it community wiki. I'd like to know the status of the question, and if it is still open, perhaps gather some new ideas for attacking it.

An automorphism of a $II_{1}$ factor is called approximately inner if it is the pointwise strong operator limit of inner automorphisms.

Let $M$ be a type $II_{1}$ factor, and let $M\overline{\otimes}M$ denote its tensor square.

Question: If the flip automorphism $\phi(x \otimes y)=y \otimes x$ of $M\overline{\otimes}M$ is path connected to the identity automorphism (in $Aut(M\overline{\otimes}M)$ with the pointwise strong operator topology), does it follow that the flip is approximately inner?