The range conjecture for the Dedekind-sum auxiliary function $\lambda$

For an integer $q\geq 2$, let $\Lambda_q$ be the set of residue classes produced by the auxiliary function $\lambda(a,t,t^{*})$, and define

$$\Gamma_q=\begin{cases}\\{24s:s\in\mathbb Z/(q^2-1)\mathbb Z\\}&q\text{ is a square},\\\\\\{12s:s\in\mathbb Z/(q^2-1)\mathbb Z\\}&\text{otherwise.}\end{cases}$$

The range conjecture. For every integer $q\geq 2$, $\Lambda_q=\Gamma_q$. This conjecture specifies the full range of the auxiliary residue-valued function and would imply Girstmair's numerator restriction conjecture. It is open in general, although the paper proves the corresponding assertions computationally for $2\leq q\leq 200$ and in several infinite families.