Uniqueness of birational structures on Inoue surfaces

We prove that the natural $(\operatorname{Aff} _2($\mathbf{C}$),$\mathbf{C}$^2)$-structureonanInouesurfaceistheunique$(\operatorname{Bir}($\mathbb{P}$^2),$\mathbb{P}$^2($\mathbf{C}$))$-structure, generalizing a result of Bruno Klingler which asserts that the natural $({\mathrm{Aff}}_2(\mathbf{C}),{\mathbf{C}}^2)$-structure is the unique $({\mathrm{PGL}}_3(\mathbf{C}),{\mathbb{P}}^2(\mathbf{C}))$-structure.