Girstmair's numerator restriction conjecture for normalized Dedekind sums

Let $S(a,b)$ be the normalized Dedekind sum, and write a value in lowest terms as $S(a,b)=k/q$, where $q\geq 2$ and $\gcd(k,q)=1$. Girstmair's necessary conditions are that $3\mid k$ when $3\nmid q$, and, when $2\nmid q$, that

$$k\equiv\begin{cases}2\pmod{4} & q\equiv 3\pmod{4},\\\\0\pmod{8} & q\text{ is a square},\\\\0\pmod{4} & \text{otherwise.}\end{cases}$$

Girstmair's conjecture. For integers $k,q$ with $q\geq 2$ and $\gcd(k,q)=1$, there exist coprime integers $a,b$ with $b\geq 1$ and $S(a,b)=k/q$ if and only if these conditions hold. The conjecture describes exactly which rational numbers with prescribed denominator occur as normalized Dedekind sums; the paper verifies it for even $q$, certain square denominators, and $2\leq q\leq 200$, but the general case remains open.