Conjecture for iterated irreducibility over global function and number fields

Let $K$ be a finite extension of $\mathbb{Q}$ or $\mathbb{F}_p(t)$, where $p$ is prime. For integers $d\geq 2$, define $K\in\mathcal{N}_{d,n}^{\infty}$ to mean that $K$ admits the relevant persistently irreducible degree-$d$ polynomial iteration property. Global-field iterates conjecture. Then

$$K\in\mathcal{N}_{2,3}^{\infty}$$

and

$$K\in\mathcal{N}_{d,2}^{\infty}\quad\text{for every }d\geq 2.$$

The conjecture isolates uniform persistent irreducibility claims for finite extensions of rational number fields and rational function fields; the paper's results motivate these cases, but their general validity is not established in the supplied text.