The global epsilon constant conjecture

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $T\Omega^{\rm loc}(L/K,1)$ be the element of $K_0(\mathbb{Z}[G],\mathbb{Q})$ constructed from the equivariant epsilon constant, the equivariant discriminant, local correction terms, and refined Euler characteristics. Global epsilon constant conjecture. For every finite Galois extension $L/K$ of number fields,

$$T\Omega^{\rm loc}(L/K,1)=0$$

in $K_0(\mathbb{Z}[G],\mathbb{Q})$. This conjecture is denoted by ${\rm EPS}(L/K)$. The conjecture gives a $K$-theoretic formulation of the vanishing of the global epsilon-constant obstruction and implies Chinburg's $\Omega(2)$-conjecture. It is known for tamely ramified extensions and for abelian extensions of $\mathbb{Q}$; the cited results also establish related equivalences with cases of the equivariant Tamagawa number conjecture under Leopoldt's conjecture.