Bogomolov–Fu–Tschinkel conjecture on torsion-coordinate intersections

Let $E_1$ and $E_2$ be elliptic curves defined over $\mathbb{C}$, given by Weierstrass equations

$$y^2=f_1(x),\qquad y^2=f_2(x),$$

where $f_1$ and $f_2$ do not have the same roots. Write $E_i(\mathbb{C})_{\rm tor}$ for the torsion points of $E_i(\mathbb{C})$. Bogomolov–Fu–Tschinkel conjecture. There is a constant $c$ such that

$$\#\bigl(x(E_1(\mathbb{C})_{\rm tor})\cap x(E_2(\mathbb{C})_{\rm tor})\bigr)\leq c.$$

The conjecture was fully proved following an initial breakthrough and the proof of the Uniform Manin–Mumford conjecture.