The equality $\psi_{\|\cdot\|_{{\mathcal{L}}}}=\psi$ holds. Hence $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$ and the metric $\|\cdot\|_{\psi}$ associated to $\psi$ by Proposition 5.16 agrees with $\|\cdot\|_{{\mathcal{L}}}$.