Second-iterate irreducibility conjecture for quadratic polynomials over the Gaussian rationals

Let $\mathbb{Z}[i]$ be the Gaussian integers and let $K$ be the field over which irreducibility and stability are considered. For $c\in\mathbb{Z}[i]\backslash\{0,-1\}$, set $r=1/c$ and $f_r(x)=x^2+r$. The polynomial $f_r$ is stable over $K$ when $f_r^n(x)$ is irreducible over $K$ for every $n\ge1$.

Second-iterate irreducibility conjecture. If $f_r^2(x)$ is irreducible over $K$, then $f_r$ is stable over $K$.

This is presented as the $\mathbb{Z}[i]$ analogue of an earlier conjecture over $\mathbb{Q}$. It asserts that irreducibility of the second iterate prevents reducibility at every later iterate, but no resolution is supplied here.