Non-emptiness conjecture for fields with persistently irreducible polynomial iterates

For a field $K$, let $K\in\mathcal{N}_{d,n}^{\infty}$ mean that there exists a degree-$d$ polynomial over $K$ whose first $n$ iterates, and all further iterates, remain irreducible. Non-emptiness conjecture. For each $d,n\geq 2$, the set $\mathcal{N}_{d,n}^{\infty}$ is non-empty. This predicts the existence of fields supporting persistently irreducible polynomial iteration in every pair of degrees and iteration lengths; the surrounding results establish several cases, while the general assertion remains open.