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 \cite{cs} 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.