@JohnRognes Thank you for a response. I have attended a talk about it, and I remember thinking that it would be hard to show that the manifolds they construct have any particular $B$-structure, because Kreck-Zagier work only rationally. But I'll check my notes to see.

@MichaelAlbanese Thank you for an answer. Yes, it is connected to the Wu class in a sense that $v_k^2=w_{2k}$ in the manifold. I will edit the question because this seems like a useful point to consider $p^*w_{2k}$.