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EDIT: This is not an answer to the question as clarified by the OP. Namely, the question imposes a condition on the homeomorphism for some $x$, but this mistaken answer assumes the condition for all $x$.

For $n \neq 1$, the given homeomorphisms act trivially on the fundamental group, but applying an orientation-reversing homeomorphism to the $S^1$-factor yields a homeomorphism that is not trivial on the fundamental group.

For $n \neq 1$, the given homeomorphisms act trivially on the fundamental group, but applying an orientation-reversing homeomorphism to the $S^1$-factor yields a homeomorphism that is not trivial on the fundamental group.

For $n \neq 1$, the given homeomorphisms act trivially on the fundamental group, but applying an orientation-reversing homeomorphism to the $S^1$-factor yields a homeomorphism that is not trivial on the fundamental group.

EDIT: This is not an answer to the question as clarified by the OP. Namely, the question imposes a condition on the homeomorphism for some $x$, but this mistaken answer assumes the condition for all $x$.

For $n \neq 1$, the given homeomorphisms act trivially on the fundamental group, but applying an orientation-reversing homeomorphism to the $S^1$-factor yields a homeomorphism that is not trivial on the fundamental group.

This problem is not research level. For $n \neq 1$, the given homeomorphisms act trivially on the fundamental group, but applying an orientation-reversing homeomorphism to the $S^1$-factor yields a homeomorphism that is not trivial on the fundamental group.

For $n \neq 1$, the given homeomorphisms act trivially on the fundamental group, but applying an orientation-reversing homeomorphism to the $S^1$-factor yields a homeomorphism that is not trivial on the fundamental group.