Bogomolov–Tschinkel uniform common-torsion conjecture for elliptic curves

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{C}$. For each $i$, let $\pi_i:E_i\to\mathbb{P}^1$ be a standard projection, meaning a degree-two quotient identifying each point $P$ with its inverse $-P$. Write $E_i^{\mathit{tors}}$ for the torsion subgroup and $E_i[2]$ for the subgroup of points killed by $2$. Assume that

$$\pi_1(E_1[2])\ne\pi_2(E_2[2]).$$

Bogomolov–Tschinkel's uniform common-torsion conjecture. There exists a uniform constant $B$ such that

$$\left|\pi_1(E_1^{\mathit{tors}})\cap\pi_2(E_2^{\mathit{tors}})\right|\leq B$$

for every such pair of elliptic curves and standard projections. This predicts uniformity in the number of common projected torsion points, independent of the curves and projections; the paper obtains its uniform Manin–Mumford theorem from a case of this conjecture, while the conjecture itself is presented as an open problem.