Semi-linearity conjecture for recurrence sets of rational maps

Let $f,g\in\mathbb{C}(z)\setminus\mathbb{C}$ be such that $\deg(f)\geq2$ or $\deg(g)\geq2$, and let $c\in\mathbb{C}(z)$. Define

$$\hat{S}_{f,g,c}=\left\lbrace n\in\mathbb{Z}_{\geq0}:\exists\lambda\in\mathbb{P}^1(\mathbb{C}),\\ f^{\circ n}(\lambda)=g^{\circ n}(\lambda)=c(\lambda)\right\rbrace.$$

Also let $\hat{S}_{f,g,c}^2$ denote the corresponding rank-two recurrence set in $\mathbb{Z}_{\geq0}^2$. The recurrence-set semi-linearity conjecture. The set $\hat{S}_{f,g,c}$ is semi-linear in $\mathbb{Z}_{\geq0}$, and $\hat{S}_{f,g,c}^2$ is semi-linear in $\mathbb{Z}_{\geq0}^2$. This extends the proved rank-one results for several non-exceptional and special exceptional cases; the general rational-map statement remains open.