The third-to-fourth iterate irreducibility conjecture for quadratic polynomials

Let $\phi_c(x)=x^2+c$ for $c\in\mathbb{Z}$, and let $b\in\mathbb{Z}$ be any basepoint. The polynomial $\phi_c^n(x)$ denotes the $n$th iterate of $\phi_c$. Third-to-fourth iterate irreducibility conjecture. The polynomial $\phi_c^3(x)-b$ is irreducible if and only if the polynomial $\phi_c^4(x)-b$ is irreducible.

Irreducibility of the third iterate implies irreducibility of the fourth iterate in all examples known in the paper, and the stated equivalence is supported by an exhaustive computational search in a bounded range. Whether it holds for all integer pairs $(b,c)$ remains open in the supplied text.