This is an important open problem, see a discussion of some of the difficulties in [Sec 1.4][1] of the book "The Ricci flow in Riemannian Geometry" of Andrews and Hopper.

As an unsolicited remark, I like better the shorter way (equivalent, by Micallef-Moore) of phrasing the question as **"Does there exist an exotic sphere with positive isotropic curvature?"**


  [1]: https://books.google.com/books?id=T_iLQ8w_N_0C&lpg=PP1&pg=PA8#v=onepage&q&f=false