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