This is an important open problem, see a discussion of some of the difficulties in Sec 1.4 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?"

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

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