Closure of independent defect under composition of valuations

Ascertain whether the composition v = v̄ ◦ w of valuations on a henselian valued field K must yield an independent defect field when both the coarsening (K,w) and the induced valuation on the residue field (Kw, v̄) are independent defect fields. Here, v̄ is a valuation on Kw and v is the composed valuation on K whose valuation ring is the preimage of the valuation ring of v̄ under the residue map of w.

Background

Independent defect is a refinement of the defect notion for Galois extensions of degree p in positive residue characteristic, where the defect set ΣL is characterized by a final segment determined by a convex subgroup H of the value group. The paper develops tools to build definable henselian valuations from independent defect extensions.

Lemma 4.2.5 establishes that if v = v̄ ◦ w with (K,w) defectless and (Kw, v̄) an independent defect field, then (K,v) is an independent defect field. The open question asks whether this property persists when both components (K,w) and (Kw, v̄) are independent defect fields, without assuming that (K,w) is defectless.