Jerónimo-Castro–Magazinov–Soberón piercing conjecture for translates of convex bodies

Let $n\ge 2$, and let $\mathcal{F}_1,\mathcal{F}_2,\dots,\mathcal{F}_n$ be families of translates of a convex compact set $\mathcal{K}$ in the plane. Assume that

$$A\cap B\neq\emptyset$$

for every $A\in\mathcal{F}_i$ and $B\in\mathcal{F}_j$ with $i\neq j$. The Jerónimo-Castro–Magazinov–Soberón conjecture. There is an index $j$ such that $\bigcup_{i\neq j}\mathcal{F}_i$ has piercing number at most $3$. This conjecture strengthens Dol'nikov's problem by allowing any number of families. The statement is known when $\mathcal{K}$ is centrally symmetric or a triangle; the source studies the problem and proves a weaker bound of $4$ points, so the bound $3$ remains open in the stated generality.