The Weak Hopf Conjecture for nonnegatively curved metrics on S^n times R^k

Let $S^n\times\mathbb{R}^k$ carry a complete metric of nonnegative sectional curvature, with soul and a small metric tube about the soul. Weak Hopf Conjecture. There does not exist a metric of nonnegative curvature on $S^n\times\mathbb{R}^k$ for which the boundary of a small metric tube about the soul has positive curvature in the induced metric. This is presented as a more accessible problem than the full Hopf conjecture and concerns the possible induced curvature on boundaries of metric tubes about souls.