Stable-factorization conjecture for quadratic polynomials over the Gaussian rationals

Let $\mathbb{Z}[i]$ be the Gaussian integers and $\mathbb{Q}(i)$ their fraction field. For $c\in\mathbb{Z}[i]\backslash\{0,-1\}$, set $r=1/c$ and $f_r(x)=x^2+r$. Let $k_n$ be the number of irreducible factors of $f_r^n(x)$ over $\mathbb{Q}(i)$.

Stable-factorization conjecture. The polynomial $f_r$ is eventually stable over $\mathbb{Q}(i)$ with constant $C(f_r,0)=4$, and exactly one of the following cases holds:

1. If $c=\alpha^2$ with $1\pm i\alpha\in\mathbb{Z}[i]\backslash\mathbb{Z}[i]^2$, then $k_n=2$ for all $n\ge 1$.
2. If $c\in\{\pm 8i,-16\}\subset\mathbb{Z}[i]^2$, then $k_1=k_2=2$ and $k_n=3$ for all $n\ge 3$.
3. If $c=(i\sigma^2-i)^2$ for $\pm\sigma\in\mathbb{Z}[i]\backslash\{3,5,56\}$, then $k_1=2$ and $k_n=3$ for all $n\ge 2$.
4. If $c=(is^2-i)^2$ for $\pm s\in\{3,5,56\}$, then $k_1=2$, $k_2=3$, and $k_n=4$ for all $n\ge 3$.
5. If $c\ne48$ and $c=\alpha^2(2i-\alpha^2)$ for some $\alpha\in\mathbb{Z}[i]$, then $k_1=1$ and $k_n=2$ for all $n\ge 2$.
6. If $c=48$, then $k_1=1$, $k_2=2$, and $k_n=3$ for all $n\ge 3$.
7. If $c\in\mathbb{Z}[i]\backslash\mathbb{Z}[i]^2$ and $c\ne\alpha^2(2i-\alpha^2)$ for every $\alpha\in\mathbb{Z}[i]$, then $k_n=1$ for all $n\ge1$.