Divisibility conjecture for an elliptic Dedekind sum

Let $a$ be an even integer and $b$ an odd integer, and let $s_{\sqrt{-1}}(a;b)$ and $Q(a;b)$ denote the elliptic Dedekind sum and the associated quantity used in the paper. For an integer $m$, consider the moduli $6m\pm1$. Divisibility conjecture. If $a$ is even and $b$ is odd, then

$$4b s_{\sqrt{-1}}(a;b)=bQ(a;b)\in4\mathbb{Z}.$$

More precisely,

$$(6m\pm1)Q(a;6m\pm1)\in12\mathbb{Z}.$$

This conjecture seeks a more precise divisibility result than the preceding lemma and is motivated by the tabulated values of $4b s_{\sqrt{-1}}(a;b)=bQ(a;b)$. Its resolution is not given in the supplied text.