Defectlessness of henselian valuations in elementary equivalents of divisible-tame type fields

Determine whether, for any non-henselian, t-henselian field K of divisible-tame type of characteristic p that is not separably closed, every elementary equivalent field L ≡ K that admits a non-trivial henselian valuation v satisfies that the valued field (L,v) is defectless.

Background

A field is of divisible-tame type if some elementary equivalent model admits a non-trivial tame valuation with divisible value group. The paper constructs non-henselian, t-henselian fields exhibiting various behaviors regarding defect and definability of valuations, and uses them to analyze the necessity of conditions in the main characterization theorem.

This question seeks a defectless analogue of Proposition 6.0.2. An affirmative answer would imply examples where none of the six conditions in the main theorem hold, showing non-existence of non-trivial definable henselian valuations in certain mixed constructions.