The bipartite Erdős–Hajnal conjecture for separately k-intersecting curve families

A family of curves is $k$-intersecting if any two curves in the family intersect in at most $k$ points. Let $\mathcal{A}$ and $\mathcal{B}$ be two families of $n$ curves each.

Bipartite curve-family conjecture. For every $k$ there is a constant $c_k>0$ such that there are subfamilies $\mathcal{A}_0\subseteq\mathcal{A}$ and $\mathcal{B}_0\subseteq\mathcal{B}$ with

$$|\mathcal{A}_0|=|\mathcal{B}_0|\ge c_k n,$$

such that either every $\alpha\in\mathcal{A}_0$ intersects every $\beta\in\mathcal{B}_0$, or every $\alpha\in\mathcal{A}_0$ is disjoint from every $\beta\in\mathcal{B}_0$.

The surrounding discussion shows that this would strengthen known results for intersection graphs of $k$-intersecting curves by requiring only the two families separately to be $k$-intersecting. It is open in the paper's context.