Is it true that any simply connected compact $n$-dimensional Riemannian manifold with positive isotropic curvature is diffeomorphic to the standard sphere $S^n$? I know that it is true for the case $n=4$ (R. Hamilton proved it in the 1990s), how about $n\geq 5$ ?

I also see that it is true for $n\geq 4$ under a stronger curvature assumption i.e. $(M,g)\times R$ has positive isotropic curvature (S. Brendle proved it in the 2000s).