The non-negative curvature and Seifert fibration conjecture for 2-connected 7-manifolds

Let $M$ be a $2$-connected $7$-manifold whose cohomology ring is that of an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$. A non-negative curvature and Seifert fibration conjecture asserts that $M$ admits a non-negatively curved, codimension-one singular Riemannian foliation with singular leaves of codimension two, and a Seifert fibration onto an orbifold $\mathbf{S}^4$ with generic fibre $\mathbf{S}^3$. This would extend the geometric constructions discussed in the paper beyond the manifolds $M^7_{\underline{a},\underline{b}}$ and would provide such structures for all 2-connected 7-manifolds with this cohomology type; the conjecture is presented as an open question.