The definable-isomorphism conjecture for infinite fields interpretable in p-adic fields

Fix a prime $p$ and let $K$ be a $(2^{\aleph_0})^+$-saturated elementary extension of $\mathbb{Q}_p$. An infinite field is interpretable in $K$ if its underlying set and field operations are interpretable in $K$.

Definable-isomorphism conjecture. Any infinite field interpretable in $K$ is $K$-definably isomorphic to some finite extension of $K$.

This folklore conjecture would imply that the structure induced on the set of balls in the valuation ring of $K$ does not interpret an infinite field, despite its Shelah completion interpreting $\mathbb{Q}_p$. Its analogue for algebraically closed valued fields is known, but the stated p-adic version is beyond current techniques.