Anti-symplectic Frey–Mazur conjecture

Let $E$ and $E'$ be elliptic curves over $\mathbb{Q}$, let $p$ be a prime, and let $E[p]$ and $E'[p]$ denote their $p$-torsion modules with Galois action by $G_{\mathbb{Q}}$. An isomorphism of these modules is anti-symplectic when it reverses the Weil pairing. Anti-symplectic Frey–Mazur conjecture. There is a constant $C^-$ such that, if $E[p]$ and $E'[p]$ are anti-symplectically isomorphic as $G_{\mathbb{Q}}$-modules for some prime $p>C^-$, then $E$ and $E'$ are $\mathbb{Q}$-isogenous. This is the anti-symplectic variant of the Frey–Mazur conjecture and remains open; in particular, the source notes that an analogous infinite family with anti-symplectically isomorphic $11$-torsion is unknown.