The colliding orbits conjecture for non-additive polynomials

Let $L$ be a field of characteristic $p>0$, let $\alpha_1,\alpha_2,\beta\in L$, and let $f\in\overline{\mathbb{F}}_p[x]$ be a polynomial of degree $d\geq 2$ that is not additive. Assume that not all of $\alpha_1$, $\alpha_2$, and $\beta$ belong to $\overline{\mathbb{F}}_p$. Define

$$C_f(\alpha_1,\alpha_2;\beta):=\{\lambda\in\overline{L}:\text{ there are }m,n\in\mathbb{N}\text{ such that }f_\lambda^m(\alpha_1)=f_\lambda^n(\alpha_2)=\beta\}.$$

The colliding orbits conjecture. The set $C_f(\alpha_1,\alpha_2;\beta)$ is infinite if and only if $f(\alpha_1)=f(\alpha_2)$. This problem concerns when the strict forward orbits of two points under the family $f_\lambda$ both contain a target point for infinitely many parameters. The monomial case $f(x)=x^d$ with $d\geq 2$ was settled, while the stated general case is presented as a conjecture.