Can you prove or disprove the following claim:

First, define the function $\xi(n)$ as follows: $$\xi(n)=\begin{cases}-1, & \text{if }\varphi(n) \equiv 0 \pmod{4} \\ 1, & \text{if }\varphi(n) \equiv 2 \pmod{4} \\ 0, & \text{if otherwise } \end{cases}$$ where $\varphi(n)$ denotes Euler's totient function. Then, $$\frac{\pi^2}{72}=\displaystyle\sum_{n=1}^{\infty}\frac{\xi(n)}{n^2}$$

The SageMath cell that demonstrates this claim can be found here.