The radius of convergence of the cotangent-sum moment series

Let

$$H_k=\frac{1}{(A_1-A_0)}\lim_{b\rightarrow+\infty}\phi(b)^{-1}b^{-2k}\sum_{\substack{A_0b\leq r\leq A_1b \\ (r,b)=1}}c_0\left(\frac{r}{b}\right)^{2k},$$

where $c_0(r/b)=-\sum_{m=1}^{b-1}\frac{m}{b}\cot\left(\frac{\pi mr}{b}\right)$, $\phi$ is the Euler phi-function, and $1/2<A_0<A_1<1$. Consider the power series

$$\sum_{k\geq 0}\frac{H_k}{(2k)!}x^k.$$

The radius-of-convergence conjecture. The radius of convergence of this series is $\pi^2$. The preceding results establish convergence for a positive range of $x$ and divergence for $|x|>\pi^2$, so the exact radius is the remaining assertion. Determining it would simplify the proof of the distribution of the cotangent sums by enabling the method of moments.