Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \cong Z$ as well? That is, does there exist a homeomorphism that extends to the boundaries as well? If not, can we at least say that $\partial Y \cong \partial Z$ even if perhaps they are "attached" to their respective interiors differently?

I rarely work with manifolds with boundary so I apologize if this is a stupid question. It seems like it should be true but I haven't been able to find a reference for this. It is certainly obvious that we have a homotopy equivalence $Y \simeq Z$ since a manifold with boundary is homotopy equivalent to its interior.