Totally ramified subfield conjecture for p-algebras

Let $K$ be a complete discrete valued field of characteristic $p>0$ with residue field $k$, and let $A$ be a $p$-algebra over $K$, meaning a central simple algebra whose center has characteristic $p$ and whose index is a power of $p$. A subfield of $A$ is totally ramified when its residue-field extension is trivial and its ramification index equals its degree over $K$. Totally ramified subfield conjecture. The algebra $A$ contains a totally ramified cyclic maximal subfield if and only if it contains a totally ramified purely inseparable maximal subfield. The paper proves this conjecture in several cases; the general equivalence remains open.