Isometric embeddings of Teichmüller spaces don't extend to the Gardiner-Masur compactification

It is known that a finitely unbranched covering $\pi :\tilde{S}\to S$ between two closed Riemann surfaces induces an isometric embedding ${\varphi }_{\pi }:T\left(S\right)\to T\left(\tilde{S}\right)$ between the corresponding Teichmüller spaces. We prove that if π is not a homeomorphism, then the induced isometric embedding ${\varphi }_{\pi }:T\left(S\right)\to T\left(\tilde{S}\right)$ doesn't extend continuously to the Gardiner-Masur compactification of the Teichmüller space, in contrast to the case of Thurston compactification and the case of augmented Teichmüller space.