Local classification of chsc Kähler metrics with cone singularities

Let $B\subset {\mathbb{C}}^n$ be a ball centred at the origin and $D=\{z^1=0\}$. Let $g$ be a Kähler metric of constant holomorphic sectional curvature (chsc) on $B\setminus D$, uniformly equivalent to the model cone metric of angle $2\pi \beta$, with $0<\beta <1$, and polyhomogeneous along $D$. We prove that, in suitable holomorphic coordinates $(w^1,\dots ,w^n)$ centred at the origin, the metric $g$ is the pullback of the corresponding complex space form by the map $(w^1,w^2,\dots ,w^n)⟼((w^1{)}^{\beta },w^2,\dots ,w^n)$.