Smoothability of the 8-dimensional Poincaré duality complex J^8

Determine whether the 8-dimensional Poincaré duality complex J^8 constructed in the paper admits a smooth structure; that is, show or refute that J^8 is smoothable.

Background

The paper constructs an 8-dimensional simply connected Poincaré duality complex J^8 whose mod 2 cohomology, as an unstable algebra over the Steenrod algebra, is F2[u2,u3]/(u2^3+u3^2, u2^2u3) with Sq^1 u2 = u3, and shows using obstruction theory that its Spivak normal bundle lifts to a PL-bundle.

Despite establishing a PL-structure and identifying a candidate stable vector bundle via calculations of [X, BSO], the authors do not prove that J^8 can be endowed with a smooth structure. Resolving smoothability would clarify whether the constructed PL Poincaré duality complex is realizable as a smooth manifold.