I sketch the proof of Buoncristiano and Hacon:

Let M be a parallelizable manifold of dimension m. Let N be M \times M \setminus U, where U is a tubular neighbourhood of the diagonal (invariant under the natural involution on M\times M.) The involution on N can be induced from the antipodal involution on the sphere S^q for a sufficiently big . Since the boundary of N is M \times S^{m-1}, hence the involution on the boundary can be induced from that on S^{m-1}.

So factorizing out by the involution the manifold N we get a manifold N', its map f to RP^q, and the boundary of N' is mapped into RP^{m-1} \subset RP^q. Take an RP^{q-m+1} in RP^q that intersects RP^{m-1} in a single point. If both f and its restriction to the boundary are  transverse to this RP^{m-1} (what can be supposed) then f^{-1}(RP^{q-m+1}) is a manifold with boundary M. Q.E.D.