The topological compactification again can be drawn from [Gro01, Prop. 2.9]. All we need to show is that a GH solution with asymptotics determined by (16) produces the right complex structure on $M$, which would then uniquely extend to $\bar{M}$ by the orbifold version of Hartog’s theorem. But this is a purely local question and it follows directly from Lemma 2.2 dropping the completeness condition (12) that becomes irrelevant.

To see matching of the volume forms let us choose local complex coordinates $\{\tilde{\eta}_{1},\dots,\tilde{\eta}_{l}\}$ on $(\mathbb{C}^{*})^{l}$ such that the local equation for $\{P_{\sigma}=0\}$ is $\tilde{\eta}_{1}=0$. In these coordinates the top degree holomorphic form on $M$ will be $\Omega=\Omega_{\tau}\wedge d\eta_{2}\wedge\dots\wedge\eta_{l}$, where $\Omega_{\tau}$ is the standard orbifold volume form. Then, $\Omega$ is easily seen to coincide with the local expression for the distinguished form $\Omega_{CY}$ on $Z_{\sigma,\tau}$. ∎