# Is a compact connected manifold-with-boundary a CW complex?

Suppose $$M$$ is a compact connected manifold-with-boundary with non-empty boundary. What can be said on whether $$M$$ can be given a CW complex structure?

A similar problem has been discussed for compact connected manifolds (i.e. with empty boundary) here, where it is true for $$\dim \neq 4$$ and not known for $$\dim = 4$$. Suppose the case for $$\dim = 4$$ is also shown true someday. Can the case with non-empty boundary be reduced to the case with empty boundary; i.e. are these problems equivalent?