Problem: Let $\gcd(a,b,c,d)$ refer to the largest integer $r$ such that $r$ divides each of $a,b,c,d$. Evaluate the series $$\sum_{a=1}^{\infty}\sum_{b=1}^{\infty}\sum_{c=1}^{\infty}\sum_{d=1}^{\infty}\frac{\mu(a)\mu(b)\mu(c)\mu(d)}{a^{2}b^{2}c^{2}d^{2}}\gcd(a,b,c,d)^{4},$$ where $\mu(n)$ is the Möbius function.
I tried several tricks, but I eventually got stuck. I think it should be possible to rewrite the entire thing as an Euler Product. It looks very similar to the double series $$\sum_{a=1}^{\infty}\sum_{b=1}^{\infty}\frac{\mu(a)\mu(b)}{a^{2}b^{2}}\gcd(a,b)^{2}=\frac{6}{\pi^2}.$$
Any help is appreciated.