Is $H_n(\mathbb{D}_\lambda^n,\mathbb{S}_\lambda^{n-1}) \simeq H_n(\overline{e_{\lambda}^n},\overline{e_{\lambda}^n}\setminus e_\lambda^n)?$

Question

In Tom Dieck p$.301$ there's the following Proposition

My problem is how to determine the inverse of such map which should be described in observation after the theorem.

In particular, I do understand that by excision we have that the inclusion $p^e$ induces an isomorphism but I don't understand how to obtain the isomorphism $\phi_*^e$ cited.

Is it true that $H_n(\mathbb{D}_\lambda^n,\mathbb{S}_\lambda^{n-1}) \simeq H_n(\overline{e_{\lambda}^n},\overline{e_{\lambda}^n}\setminus e_\lambda^n)?$

Doesn't seem so. Also, the isomorphism should be quite easy since the composition should be the inverse of $\phi^n$, which I'm unable to show.

Any suggestion answer or reference would be appreciated.