Let $M^2,N^2$ be connected closed surfaces. Suppose there exists region $D$ in the interior of $M times (2,2)$ such that (a) $D$ is homeomorphic to $N times (0,1)$; (b) $D$ contains $M times (1,1)$.
Can we prove the following statements?

$M$ and $N$ are homotopic.

$M$ and $N$ are homeomorphic.
If true, are there any generalizations for higher dimensional case?