On the inverse stability of $z^n+c\mathrm{.}$

Let $K$ be a field and $\phi (z)\in K[z]$ be a polynomial. Define $\mathrm{\Phi }(z):=\frac{1}{\phi (z)}\in K(z)\mathrm{.}$ For $n\in {\mathbb{N}}^{\ast }$, let the $n$-th iterate of $\mathrm{\Phi }(z)$ be defined as ${\mathrm{\Phi }}^{(n)}(z)=\underset{n\text{ times}}{\underbrace{\mathrm{\Phi }\circ \mathrm{\Phi }\circ \cdots \circ \mathrm{\Phi }}}(z)\mathrm{.}$ We express the ${\mathrm{\Phi }}^{(n)}(z)$ in its reduced form as ${\mathrm{\Phi }}^{(n)}(z)=\frac{f_{n,\phi }(z)}{g_{n,\phi }(z)},$ where $f_{n,\phi }(z)$ and $g_{n,\phi }(z)$ are coprime polynomials in $K[z]$. A polynomial $\phi (z)\in K[z]$ is called inversely stable over $K$ if every $g_{n,\phi }(z)$ in the sequence $\{g_{n,\phi }(z){\}}_{n=1}^{\mathrm{\infty }}$ is irreducible in $K[z]$. This paper investigates the inverse stability of the binomials $\phi (z)=z^d+c$ over $K$.