  In a recent work the authors prove the effective asymptotic Fermat's Last
Theorem for the infinite family of fields $\mathbb{Q}(\zeta_{2^{r+2}})^+$ where
$r \ge 0$. A crucial step in their proof is the following conjecture of Kraus.
Let $K$ be a number field having odd narrow class number and a unique prime
$\lambda$ above $2$. Then there are no elliptic curves defined over $K$ with
conductor $\lambda$ and a $K$-rational point of order $2$. In this note we give
a new elementary proof of Kraus' conjecture that makes use only of basic facts
about elliptic curves, Tate curves and Tate modules.
