The volume-form preservation conjecture for automorphisms of $(\mathbb{C}^*)^2$

Let $F\in\operatorname{Aut}(\mathbb{C}^*\times\mathbb{C}^*)$. The holomorphic $2$-form on $(\mathbb{C}^*)^2$ is

$$\frac{dz\wedge dw}{zw}.$$

Volume-form preservation conjecture. Every automorphism $F$ of $\mathbb{C}^*\times\mathbb{C}^*$ preserves this form:

$$F^*\left(\frac{dz\wedge dw}{zw}\right)=\frac{dz\wedge dw}{zw}.$$

This conjecture is the automorphism statement assumed in the paper's main result. Its resolution status is not specified in the supplied text.