Conjecture on uniqueness of rational maps from a numerator of a 3-cycle

Let $A(m,n)$ and $B(m,n)$ be the integer forms parametrizing rational $3$-cycles of maps $f_c(x)=x^2+c$, and let $(m,n)\in\mathbb{Z}^2$ satisfy $mn(m+n)\neq0$ and $\gcd(m,n)=1$. Numerator-uniqueness conjecture. If $A(m,n)=k$, then $c=-A(m,n)/B(m,n)$ is the only rational number with numerator $k$ for which $f_c$ has a rational $3$-cycle. Equivalently, for $k\in\mathbb{N}$, the equation

$$A(m,n)=m^6+2m^5n+4m^4n^2+8m^3n^3+9m^2n^4+4mn^5+n^6=k$$

has either $0$ or $6$ solutions $(m,n)$ satisfying $mn(m+n)\neq0$ and $\gcd(m,n)=1$. The conjecture was checked computationally for $k\leq10^{15}$; proving it amounts to establishing uniqueness of the relevant norm representations beyond the six transformations described in the paper.