The average word-length conjecture for trace classes

Let $L$ be the ambient subgroup, let $h(t)$ count the $\operatorname{SL}_2(\mathbb{Z})$-conjugacy classes of hyperbolic matrices in $L$ with trace $t$, and let $\ell([g])$ denote the word length of a conjugacy class representative. The average word-length conjecture. As admissible $t$ tends to infinity,

$$\frac{1}{h(t)}\sum_{\substack{[g]\subset L\operatorname{Tr}(g)=t}}\ell([g])\asymp\log(|t|).$$

The conjecture formalizes the observed logarithmic average word length and supports the class-number heuristic for $\Gamma(2)'$; the source supplies computational evidence rather than a proof.