Extension of Theorem 1.2 to the spin case remains unclear

Establish whether the diffeomorphism, homeomorphism, and homotopy classification results of Theorem 1.2—formulated for nonspin 13-manifolds M with H*(M; Z) ≅ H*(CP^3 × S^7; Z) that admit restriction lifts—also hold for simply connected, closed, smooth 13-dimensional spin manifolds with the same cohomology ring; equivalently, determine if an analogue of Theorem 1.2 is valid in the spin case.

Background

Theorem 1.2 provides classification criteria for nonspin 13-manifolds M with H*(M; Z) ≅ H*(CP^3 × S^7; Z) that admit restriction lifts, characterizing diffeomorphism, homeomorphism, and homotopy equivalence in terms of the first Pontrjagin class p1(M) and its congruences.

The author notes that in a special spin family, CP^3 × Σ^7 (with Σ^7 a homotopy 7-sphere), the manifolds are diffeomorphic to CP^3 × S^7, but indicates that extending the full classification results analogous to Theorem 1.2 to the general spin case is presently unclear.