On the cobordism groups of $O\langle n\rangle$-manifolds

Given a smooth manifold, a tangential $O\langle n \rangle$-structure is a lift of the classifying map of its tangent bundle to $BO\langle n \rangle$. An $O\langle n \rangle$-manifold is a manifold equipped with an equivalence class of tangential $O\langle n \rangle$-structures. The cases $n=2$ and $