The two-limit conjecture for the $(1,2)$ and $(2,3)$ games

Fix $(p,q)=(1,2)$ or $(p,q)=(2,3)$, and let $(a_n,b_n)$ be the associated sequences. Two sequences $(l_i)$ and $(u_i)$ are complementary when they are increasing and together partition the relevant index set, as defined in the paper. The two-limit conjecture. There exist increasing complementary sequences $(l_i)$ and $(u_i)$ such that both limits

$$\eta=\lim_{i\rightarrow\infty}\frac{b_{l_i}}{a_{l_i}},\qquad \gamma=\lim_{i\rightarrow\infty}\frac{b_{u_i}}{a_{u_i}}$$

exist and satisfy

$$1<\eta<\Phi<\gamma\leq 3.$$

This predicts two distinct asymptotic ratio regimes, separated by the golden ratio, for these two parameter choices.