On the paper Manifolds with positive sectional curvature almost everywhere
Burkhard Wilking asks the following (Question $2$ pg. 121):
Let $(M^n,g)$ be a compact Riemannian manifold with non--negative sectional curvature. Assume that there is an open subset $U\subset M$ diffeomorphic to $\mathbb{R}^n$ such that $g$ has positive sectional curvature on $M\setminus U$. Does $M$ admit a Riemannian metric of positive sectional curvature?
I would like to know if this question is already answered on the literature in its full generality or with symmetry assumptions, e.g, on the presence of an isometric and effective action by a Lie group.
