Non-negative curvature on certain product manifolds

Let $G\mathrm{/}H$ be a closed, simply connected homogeneous manifold. Suppose every stable class of real vector bundles over $G\mathrm{/}H$ contains a homogeneous bundle. Then, for any closed, simply connected smooth manifold $M$ homotopy equivalent to $G\mathrm{/}H$, there exists $n>\mathrm{d}\mathrm{i}\mathrm{m}(M)$ such that the product manifold $M\times S^n$ admits a metric with non-negative sectional curvature. Many homogeneous manifolds satisfy this assumption, including simply connected compact rank-one symmetric spaces, and among others.