Kraus's conjecture on elliptic curves over totally real fields

Let $K$ be a totally real number field with narrow class number $1$. Suppose that $2$ is totally ramified in $K$, and let $d513$ be the unique prime of $K$ above $2$.

Kraus's conjecture. There are no elliptic curves over $K$ with full $2$-torsion and conductor $d513$.

The paper states that Kraus's conjecture implies asymptotic Fermat's Last Theorem for such fields. The source also presents a strengthened theorem later, but this candidate itself is the conjecture and its resolution is not indicated by the supplied status.