Stability conjecture for irreducible polynomials of the form $z^d+1/c$

Let $d\geq 3$ be an integer, let $c\in\mathbb{Z}\setminus\{0\}$, and set $f(z)=z^d+\frac{1}{c}$. The polynomial $f$ is irreducible over $\mathbb{Q}$ when it is irreducible as a polynomial in $\mathbb{Q}[z]$, and it is stable over $\mathbb{Q}$ when every iterate $f^n(z)$ is irreducible over $\mathbb{Q}$.

Stability conjecture. If $f(z)=z^d+\frac{1}{c}$ is irreducible over $\mathbb{Q}$, then $f(z)$ is stable over $\mathbb{Q}$.

The paper proves stability in several families of degrees and notes that the claim is implied by the explicit $abc$-conjecture. The general assertion remains open.