  Let $E$ and $A$ be elliptic curves over a number field $K$. Let $\chi$ be a
quadratic character of $K$. We prove the conjecture posed by Mazur and Rubin on
$n$-Selmer near-companion curves in the case $n=2$. Namely, we show if the
difference of the $2$-Selmer ranks of $E^\chi$ and $A^\chi$ is bounded
independent of $\chi$, there is a $G_K$-isomorphism $E[2] \cong A[2]$.
