Strengthening the AKE_≡ result to pointed equivalence for e>0

Establish whether, for any prime p and imperfection degree e>0, and any pointed valued fields (K,v,t) and (L,w,s) in the class C_{p,e}, the biconditional holds: ((vK,v(t)) ≡ (wL,w(s)) and O_v/(t) ≡ O_w/(s)) if and only if (K,v,t) ≡ (L,w,s).

Background

The paper proves a relative completeness (AKE_≡-type) result: if (vK,v(t)) ≡ (wL,w(s)) and O_v/(t) ≡ O_w/(s), then (K,v) ≡ (L,w). In the perfect case (e=0), known results allow strengthening this to equivalence of the pointed structures (K,v,t) ≡ (L,w,s), using a common perfect subfield in C_{p,0}.

For positive imperfection degree e>0, it is not known whether this strengthening still holds. The authors explicitly pose this as an open question, asking whether one can upgrade equivalence of value-group-with-point and residue ring modulo the parameter to full elementary equivalence of the pointed valued fields.