Counting quaternion extensions by multi-heights

We prove a case of a conjecture of Gundlach on the number of Galois extensions over $\text{QQ}$ with bounded multi-invariants for the group $G=Q_8$. We also obtain results when imposing local conditions at the wildly ramified place. As an application, we prove asymptotics for the number of $Q_8$-extensions with bounded product of ramified primes.