Cyclic versus purely inseparable totally ramified maximal subfields of p-algebras

Let $K$ be a complete discrete valued field of characteristic $p$ with residue field $k$, and let $A$ be a $p$-algebra over $K$. A maximal subfield of $A$ is totally ramified cyclic or totally ramified purely inseparable according to the properties asserted below. Cyclic versus purely inseparable maximal subfield conjecture. $A$ contains a totally ramified cyclic maximal subfield if and only if it contains a totally ramified purely inseparable maximal subfield. The conjecture asks whether these two types of totally ramified maximal subfields always occur together; the source paper proves it under some conditions on the $p$-rank of the residue field $k$, while the unrestricted statement is presented as a conjecture.