Weak Frey–Mazur conjecture for elliptic curves over number fields

Let $K$ be a number field and let $E_0$ be a fixed elliptic curve over $K$. Let $p$ be a prime, and let $E/K$ be an elliptic curve. Write $\bar{\rho}_{E,p}$ and $\bar{\rho}_{E_0,p}$ for the residual mod-$p$ Galois representations of $E$ and $E_0$.

Weak Frey–Mazur conjecture. There is a constant $N_{E_0}$, depending only on $E_0$, such that if $p>N_{E_0}$ and

$$\bar{\rho}_{E,p}\sim \bar{\rho}_{E_0,p},$$

then $E$ is isogenous to $E_0$.

This is a weak form of the Frey–Mazur conjecture over number fields: sufficiently large-prime congruence of residual representations should force isogeny. The source attributes it to Frey and Mazur and uses it as an input; no resolution is stated here.