Spineless 5-manifolds and the deformation conjecture

We construct a compact PL 5-manifold M (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, ∨^_1 1S^2, which is "spineless", meaning M is not the regular neighborhood of any 2-complex PL embedded in M. We formulate a related question about the existence of exotic smooth structures on 4-manifolds which is of interest in relation to the deformation conjecture for 2-complexes, also known as the generalized Andrews-Curtis conjecture.