Finite-energy equivariant minimal immersions in $\mathbb{CH}^2$ with fixed-lift proper cusp ends

Let $Σ_0\,=\,Σ\setminus\{z_1,\,\ldots,\, z_n\}$ be a finite-type hyperbolic Riemann surface, with its universal cover being identified with $\mathbb H$. We prove that a finite-energy equivariant harmonic map \[ f\colon\mathbb H\;\longrightarrow\;\mathbb{CH}^2 \] is fixed-lift proper at every punctur