Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ N \neq \emptyset$ and $\bullet M \approx \bullet N$, where $\bullet M$ denotes the interior of $M$ and $\approx$ denotes homeomorphic. Does it necessarily hold that $N \approx M$?
EDIT: Since there were no replies here, I asked the same question on MathOverflow. The surprising answer is no.