Large irreducible factors in iterated polynomial-value differences

Let $\alpha,\beta\in\mathbb{F}_q$, let $\lambda\in\overline{\mathbb F}_p$, and let $f_\lambda(z)=z^d+\lambda$. Define $P_{n,\alpha}(\lambda)=f_\lambda^n(\alpha)\in\mathbb{F}_q[\lambda]$, a polynomial of degree $d^{n-1}$. Suppose $\gcd(d,q-1)=1$. Large-factor conjecture. For every $M>0$, there is an $n\in\mathbb{N}$ such that $P_{n,\alpha}(\lambda)-\beta$ has an irreducible factor $g(\lambda)\in\mathbb{F}_q[\lambda]$ of degree $k>M$ satisfying $\gcd(d,q^k-1)=1$. This prediction would provide arbitrarily large finite-field extensions containing parameters for which the relevant orbit condition is satisfied; the source supplies no resolution status.