Restrict attention to the ball $B_{2\rho_{0}}(q_{j})$. First note that $1+\frac{\epsilon\,(m_{j}-2)}{\rho}\geq 1-\frac{2\epsilon}{\rho}=\tfrac{3}{4}$ for $\rho=8\epsilon$. Now choose $\epsilon_{0}>0$ so that $\epsilon(\lambda_{j}+C\tfrac{\epsilon^{2}}{64})\leq\tfrac{1}{4}$ for all $\epsilon\leq\epsilon_{0}$. Here $\lambda_{j},C$ are the constants of Lemma 4.7.(i). We conclude that $h_{\epsilon}>\tfrac{1}{2}$ on $\partial B_{8\epsilon}(q_{j})$ for $\epsilon<\epsilon_{0}$. Since $h_{\epsilon}$ blows up to $+\infty$ at the punctures $q_{k+1},\dots,q_{8},\pm p_{1},\dots,\pm p_{n}$ the maximum principle completes the proof. ∎