The local-global conjecture for traces in the commutator subgroup of $\Gamma(2)$

Let $\Gamma(2)'$ be the commutator subgroup of the level-2 principal congruence subgroup $\Gamma(2)\subset\operatorname{SL}_2(\mathbb{Z})$, and let an integer $t$ be admissible if, for every $q\geq1$, it belongs to $\operatorname{Tr}(\Gamma(2)')\bmod q$. The local-global trace conjecture. For all sufficiently large admissible $t$, there exists $\gamma\in\Gamma(2)'$ such that $\operatorname{Tr}(\gamma)=t$. The admissibility theorem identifies the local congruence conditions, while the global assertion that every sufficiently large admissible trace occurs remains unresolved.