As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature.
But Hopf's conjecture is unsolved, i.e., the conjecture: $S^2\times S^2$ has a positive curvature.
So I think that the quasi-positive curvature-condition can be weakened by conditions : non-negative curvature and non-flatness.
Here my concrete question is : Is there a simply connected compact simple Lie group which does not have a metric with a positive curvature and has a metric with a non-negative curvature ?
Here we say that a Riemannian manifold $M$ has positive (resp. non-negative) curvature if all sectional curvatures are positive (resp. non-negative) at all points of $M$ And a manifold with quasi-positive curvature is a manifold with a non-negative curvature and a point at which all tangent 2-planes have positive curvature.