Let $M$ be a Cauchy space whose induced topological space is a second-countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^m$.

Does it follow that there exists a subspace $N$ of $\mathbb{R}^n$ and a function $f : M \to N$ such that $f$ is a Cauchy-continuous bijection whose inverse is also Cauchy-continuous?
If no, what other conditions would be sufficient/insufficient to conclude this?