The class-number asymptotic conjecture for $\Gamma(2)'$

Let $L$ be the ambient subgroup under consideration, and let $h(t)$ be the number of $\operatorname{SL}_2(\mathbb{Z})$-conjugacy classes of hyperbolic matrices in $L$ with trace $t$. Let $h'(t)$ be the corresponding number for $\Gamma(2)'$. An integer $t$ is admissible if it occurs in the trace set modulo every positive integer. The class-number conjecture. For admissible $t$,

$$h'(t)\asymp\frac{h(t)}{\log(|t|)}.$$

The conjecture models the proportion of conjugacy classes whose homology is trivial; the paper gives heuristic and computational evidence but no proof.