The compactification conjecture for subsets of Euclidean space

Let $A$ be an $m$-dimensional subset of $\mathbb{R}^n$. An $m$-dimensional compactification of $A$ is a compact space containing $A$ as a dense subspace and having dimension $m$.

Compactification conjecture. Every $m$-dimensional subset of $\mathbb{R}^n$ has an $m$-dimensional compactification that can be embedded into $\mathbb{R}^n$.

This conjecture would complete the reduction of the problem of whether covering dimension satisfies Menger's axioms. The source does not state a resolution, so the conjecture is recorded as open.