Ito's conjecture on the mean absolute value of elliptic Dedekind sums

Let $K_2(X)$ be the Farey set of elements $a/c$ in the class-number-one imaginary quadratic field with discriminant parameter $D=2$ and $|c|^2<X$, and let

$$B_2:=\{z=x+iy:0\leq x<1,\ 0<y\leq \frac{1}{\sqrt{2}}\}.$$

For the normalized elliptic Dedekind sum $\tilde{\mathfrak{D}}(a,c)$, Ito's conjecture.

$$\lim_{X\to\infty}\frac{1}{\#(K_2(X)\cap B_2)}\sum_{\frac{a}{c}\in K_2(X)\cap B_2}\left|\tilde{\mathfrak{D}}(a,c)\right|=\infty.$$

Ito formulated this conjecture from numerical experiments for $D=2$. The paper proves it as an immediate corollary of its Gaussian limiting-distribution theorem for suitably normalized Sczech sums.