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

A claim in Mochizuki [Moc15, Theorem 1.11] asserts slimness for absolute Galois groups of Kummer-faithful fields of characteristic zero. The authors report that Mochizuki indicated a gap in the proof, leaving the status of this assertion open in general.

References

If we accepted the claim in [Moc15, Theorem 1.11] that the absolute Galois group of any Kummer-faithful field of characteristic zero is slim, then the field K(o) would not be Kummer-faithful for any o E GK since the absolute Galois group GK(g) of K (o) is abelian. However, Mochizuki recently informed us that there is a gap in the proof of this claim and the status remains open for general Kummer-faithful fields of characteristic zero.

— Mordell--Weil groups over large algebraic extensions of fields of characteristic zero  (2408.03495 - Asayama et al., 2024) in Remark 5.5