Slimness of absolute Galois groups of Kummer-faithful fields of characteristic zero

Determine whether the absolute Galois group of every Kummer-faithful field of characteristic zero is slim (i.e., every open subgroup has trivial center).

Background

Mochizuki’s Theorem 1.11 in [Moc15] claimed slimness of the absolute Galois group for Kummer-faithful fields of characteristic zero, a property relevant to anabelian geometry. The present paper reports a gap in the proof and leaves the general case unresolved, even though the authors’ results show that certain fields K(σ) have abelian absolute Galois groups, contradicting the slimness claim if it were universally true.