Conjecture on longest orbit lengths for ambiguous and non-ambiguous modular elements

Let $g\in\operatorname{PSL}_{2}(\mathbb{Z})$ be hyperbolic. Let $L(g;p)$ denote the longest orbit of $g$ on $\mathbb{Y}_{-2}(\mathbb{F}_{p})$, where $\mathbb{Y}_{-2}(\mathbb{F}_{p})$ is the set of orbits of the even-sign-change Klein four-group on the nonzero Markoff surface over $\mathbb{F}_{p}$. An element $g$ is ambiguous if its conjugacy class is taken to the conjugacy class of $g^{-1}$ by an element of $\operatorname{PGL}_{2}(\mathbb{Z})$ of determinant $-1$. Longest-orbit conjecture. If $g$ is ambiguous, then there are constants $C_{1}=C_{1}(g)>0$ and $C_{2}=C_{2}(g)>C_{1}$ such that

$$C_{1}p\leq L(g;p)\leq C_{2}p$$

for all primes $p$, and the measures

$$\frac{1}{\#\{\text{primes }p\leq X\}}\left(\sum_{p\leq X}\delta_{L(g;p)/p}\right)$$

converge as $X\to\infty$ to a compactly supported Borel probability measure on $\mathbb{R}$. If $g$ is not ambiguous, then there is a constant $c=c(g)$ such that $L(g;p)\geq cp^{2}$ for all primes $p$, and the measures

$$\frac{1}{\#\{\text{primes }p\leq X\}}\left(\sum_{p\leq X}\delta_{L(g;p)/p^{2}}\right)$$

converge as $X\to\infty$ to a compactly supported Borel probability measure on $\mathbb{R}$. In particular, for $\kappa=-2$,

$$\lim_{p\to\infty}\frac{\log L(g;p)}{\log p}=\begin{cases}1&\text{if }g\text{ is ambiguous},\\2&\text{if }g\text{ is not ambiguous.}\end{cases}$$

This conjecture predicts a sharp dichotomy between linear and quadratic longest-orbit growth, together with limiting distributions after the corresponding normalization. It is motivated by computations for explicit hyperbolic words and by the equivalence between cyclic-palindrome and ambiguous elements established earlier in the paper; the asserted bounds, convergence statements, and limiting exponents remain open.