The commutator-width conjecture for traces in $\Gamma(2)'$

Let $\Gamma(2)'$ be the commutator subgroup of $\Gamma(2)$, and let $\operatorname{Tr}(\Gamma(2)')$ denote its trace set. The commutator width of an element is the minimum number of commutators needed to express it. The commutator-width conjecture. For every $t\in\operatorname{Tr}(\Gamma(2)')$, there exists $\gamma\in\Gamma(2)'$ such that $\operatorname{Tr}(\gamma)=t$ and $\gamma$ has commutator width either 1 or 2. Computational results in the paper show that some traces have no 1-commutator representative, motivating the asserted bound of 2; the general statement remains open.