The converse characterization of extreme points in $PRP_2$

Let $p\in\mathbb{C}[z_1,z_2]$ be scattering stable, let $\widetilde{p}(0,0)=0$, and define

$$f=\frac{p+\widetilde{p}}{p-\widetilde{p}}.$$

Converse characterization conjecture. If $f$ is an extreme point of $PRP_2$, then $p$ is $\mathbb{T}^2$-saturated and $p-\widetilde{p}$ is irreducible.

The preceding theorem proves that $\mathbb{T}^2$-saturation of $p$ together with irreducibility of $p-\widetilde{p}$ is sufficient for $f$ to be extreme. This conjecture asserts that these conditions are also necessary, giving a characterization of the extreme points arising from such rational functions. The source states that this remains unproved.