Instability conjecture for finite-field polynomials of degree divisible by the characteristic

Let $p$ be a prime, let ${\mathbb F}_p$ be the finite field with $p$ elements, and let $f(x)\in{\mathbb F}_p[x]$ be a polynomial whose degree is divisible by $p$. A polynomial over a field is stable if every positive iterate is irreducible over that field. Instability conjecture. The polynomial $f(x)$ is not stable over ${\mathbb F}_p$. This conjecture concerns the stability of polynomial iteration over finite fields and is motivated by computations on trinomials. The paper gives counterexamples to an earlier question predicting that the $(p+1)$-st iterate would always be reducible, while conjecturing the weaker assertion that some iterate is reducible.