The totally ramified extension conjecture for potentially good reduction

Let $K$, $\phi$, and $B$ be as in Theorem A, and let $k$ be the residue field of $K$. Assume that $k$ is perfect.

Totally ramified extension conjecture. If $\phi$ has potentially good reduction, then there is a totally ramified extension $L/K$ with $[L:K]\leq B$ such that $\phi$ is conjugate over $L$ to a map of good reduction.

The conjecture proposes that the sharp bound $B$ from Theorem A can always be attained by a totally ramified extension when the residue field is perfect. The paper notes that the perfectness hypothesis is necessary and suggests proving the conjecture through totally ramified analogues of the auxiliary theorems used for Theorem A.