The global equivariant epsilon constant conjecture

Let $F/E$ be a Galois extension with Galois group $\Gamma$, and let $T\Omega^\mathrm{loc}(F/E,1)$ be the locally defined element of $K_0(\mathbb Z[\Gamma],\mathbb R)$. Global equivariant epsilon constant conjecture. One has

$$T\Omega^\mathrm{loc}(F/E,1)=0$$

in $K_0(\mathbb Z[\Gamma],\mathbb R)$. This conjecture is equivalent to a conjecture of Bley and Burns and is known for at most tamely ramified extensions, abelian extensions of $\mathbb Q$ with odd conductor, and extensions of $\mathbb Q$ of degree at most $15$. It also implies Chinburg's $\Omega(2)$-conjecture.