The totally ramified point conjecture for disks

Let $K$, $f$, $V$, $p$, $n$, and $e$ be as in Lemma~, and let $k$ be the residue field of $K$. Assume that $k$ is perfect.

Totally ramified point conjecture. There is a point $\alpha$ in the disk $V$ such that $K(\alpha)/K$ is totally ramified and

$$[K(\alpha):K]\leq p^e.$$

The paper identifies this statement as the key extension of Lemma~ needed to establish the preceding conjecture. It remains open in the supplied text.