Let $M$ be a topological manifold. We know that $M$ is stably smoothable if and only its tangent microbundle, up to stabilization, admits a reduction to vector bundle.

Now I wonder if there is a relative version of this fact. Suppose $M$ and $N$ are topological manifolds and $f: M \to N$ is a locally flat topological embedding and assume it admits a normal microbundle. We also assume that the normal microbundle is equivalent to a vector bundle. Suppose $M$ and $N$ are both stably smoothable. I wonder if one can prove that there are smoothable stablizations $M = M \times {\mathbb R}^m$ and $N' = N \times {\mathbb R}^{m + n}$ such that the natural extension $f': M' \to N'$ (extending $f$ by the inclusion ${\mathbb R}^m \to {\mathbb R}^{m+n}$) is a smooth embedding.

If the above speculation is true, I also wonder if such stable smoothings are in any sense unique.