Totally Ramified Subfields of p-Algebras Over Discrete Valued Fields with Imperfect Residue

Let K be a complete discrete valued field of characteristic p with residue k which is not necessarily perfect. We prove the Conjecture in Chapman and Srimathy (Totally ramified subfields of p-algebras 2024) that a p-algebra over K contains a totally ramified cyclic maximal subfield if it contains a totally ramified purely inseparable maximal subfield provided k satisfies some conditions on its p-rank.