Quadratic isogenies conjecture for quadratic points on modular curves

Let $X_0(N)$ be the modular curve parametrizing cyclic isogenies of degree $N$. A point is CM if it corresponds to an elliptic curve with complex multiplication, and a cusp is a cusp of $X_0(N)$. Let $K$ be a quadratic field and let $N$ be an integer.

Quadratic isogenies conjecture. There exists an integer $C$, independent of $K$, such that if $N>C$, then every point

$$P\in X_0(N)(K)$$

is either a cusp or a CM point.

This conjecture predicts that, apart from CM points, quadratic points occur on only finitely many modular curves $X_0(N)$, uniformly over all quadratic fields. It implies Elkies's conjecture that only finitely many $N$ have a non-CM, non-cuspidal rational point on the quotient $X_0^+(N)$.