McMullen's exponential-growth conjecture for bounded continued fractions

Consider the periodic continued fractions $[\overline{a_1,\ldots,a_k}]$ with entries in $\mathbb{Q}(\sqrt{5})$ satisfying $a_i\leq 2$. McMullen's exponential-growth conjecture. The set

$$\{[\overline{a_1, \ldots, a_k}]\in \mathbb{Q}(\sqrt{5})\mid a_i\leq 2\}$$

has exponential growth as $k\to\infty$. This conjecture motivates the paper's local-global investigation of trace sets for thin semigroups.