The local-global multiplicity conjecture for traces in $\Gamma(2)'$

Let $\Gamma(2)'$ be the commutator subgroup of $\Gamma(2)$, let $t$ be an admissible trace, and define

$$\operatorname{mult}_X(t)=\#\{\gamma\in\Gamma(2)' \mid \lVert\gamma\rVert<X\text{ and }\operatorname{Tr}(\gamma)=t\}.$$

The local-global multiplicity conjecture. If $t\asymp X$ is admissible, then

$$\operatorname{mult}_X(t)>X^{1-o(1)}.$$

This is the multiplicity analogue of the existence conjecture for admissible traces; the source presents it as another version of the local-global conjecture and does not establish it.