The stable converse soul question for positively curved homogeneous spaces

The stable converse soul question (SCSQ) asks whether, given a real vector bundle $E$ over a compact manifold, some stabilization $E\times\R^k$ admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space $B^{13}$. Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact rank one symmetric spaces of dimensions multiples of four, and for certain products of spheres. Moreover, we observe that the SCSQ is "stable under tangential homotopy equivalence": if it has an affirmative answer for all vector bundles over a certain manifold $M$, then the same is true for any manifold tangentially homotopy equivalent to~$M$. Our main tool is topological K-theory. Over $B^{13}$, there is essentially one stable class of real vector bundles for which our method fails.