Finiteness conjecture for sporadic quadratic points on modular curves

For $N\in{\mathbb Z}_{\ge 1}$, let $X_0(N)$ be the modular curve classifying cyclic isogenies of degree $N$. A quadratic point is sporadic if it is not a cusp, is isolated rather than arising from a degree-$2$ hyperelliptic or bielliptic covering over a rational point, and does not classify a CM elliptic curve and cyclic isogeny of degree $N$ of the type described in the source.

Sporadic quadratic-point conjecture. Ranging over all $X_0(N)$ for $N\in{\mathbb Z}_{\ge 1}$, there are only finitely many sporadic quadratic points.

Quadratic points coming from hyperelliptic and bielliptic coverings, or from CM elliptic curves over the listed class-number-one imaginary quadratic fields, are excluded because they form expected families. The conjecture asserts finiteness of the remaining exceptional points.