Consider the pairs $(\mathbb D^{n},S^{n-1})$ and $(\mathbb D^{n},\mathbb D^{n}-\{0\})$ ,clearly their homologies are same in each dimensions but these pairs are not homotopy equivalent.
Any homotopy equivalence $f:(X,A)\to(Y,B)$ induce a homotopy equivalence $f:(X,\bar A)\to (Y,\bar B) $.
If the pairs in the questions are homotopy equivalent then the pairs $(\mathbb D^{n},S^{n-1})$ and $(\mathbb D^{n},\mathbb D^{n})$ are homotopy equivalent which is certainly not true as their homologies are different at $n$th dimension.
Any comment or discussion will be appreciated.