Framed homeomorphism implies diffeomorphism

Let $M$ and $M'$ be smooth compact manifolds smoothly embedded in euclidean space, and let the embeddings define flat framed stratifications. Framed homeomorphism implies diffeomorphism. If the resulting flat framed stratifications are flat framed homeomorphic, then $M$ and $M'$ are diffeomorphic. This conjecture seeks to extend the established correspondence between flat framed topological and piecewise linear structures to smooth topology; its status is not resolved in the source.