Strong growth conjecture for Fibonacci-polynomial cycles with m = 8 + 12(61 + 64d)

Let $m=8+12q$ with $q=61+64d$ for some nonnegative integer $d$. Define the sequence $\{g_l\}_{l\ge 10}$ as in the preceding definition, with $g_{10}$ and $g_{10}'$ selected according to the parity of $d$, and with each subsequent pair determined by whether $\{g_{l-1}',F_m(g_{l-1}')\}$ is a length-two cycle strongly growing at level $l$. Strong growth conjecture. For every $l\ge 10$, the Fibonacci polynomial $F_m(x)$ has a cycle $\{g_l,F_m(g_l)\}$ of length $2$ that strongly grows at level $l$. The authors report computational evidence for many values of $l$, but the apparently random sequence prevents a proof of strong growth and the conjecture remains open.