Infinite collision conjecture over the algebraic closure of a finite field

Let $d\ge 2$ be an integer that is not a power of $p$, and let $\alpha_1,\alpha_2,\beta\in\overline{\mathbb F}_p$. For each $\lambda\in\overline{\mathbb F}_p$, let $f_\lambda(z)=z^d+\lambda$, and define

$$C(\alpha_1,\alpha_2;\beta)=\left\{\lambda\in\overline{\mathbb F}_p: f_\lambda^m(\alpha_1)=f_\lambda^n(\alpha_2)=\beta\text{ for some }m,n\in\mathbb{N}\right\}.$$

Infinite collision conjecture. The set $C(\alpha_1,\alpha_2;\beta)$ is infinite. The conjecture reverses the expectation stated immediately beforehand, which predicted finiteness when $\alpha_1^d\ne\alpha_2^d$; the source reports extensive computation but gives no proof or resolution.