Gundlach's multi-invariant counting conjecture for Galois extensions

Let $G$ be a finite group, let $K$ be a number field, and let $\operatorname{inv}_1,\ldots,\operatorname{inv}_n$ denote the multi-invariants used to order $G$-extensions $M/K$, with $\operatorname{Gal}(M/K)\simeq G$. Fix a real number $0<\delta<1$. Gundlach's multi-invariant conjecture. There is a constant $C>0$ such that, as all $X_i$ tend to infinity,

\\#\\{M/K:\operatorname{Gal}(M/K)\simeq G,\\ \delta X_i<\operatorname{inv}_i(M)\leq X_i\\ \forall i\\}\sim C\cdot\prod_i X_i.

The conjecture is motivated by a heuristic for counting extensions in multi-invariant boxes and is intended to refine Malle's discriminant-ordering conjecture. The paper applies this framework to $D_4$-extensions over $\operatorname{\mathbb{Q}}$, but the supplied status information does not establish a resolution in general.