Dol’nikov's colorful piercing conjecture for arbitrarily many families

Let $K$ be a compact convex set in $\mathbb{R}^{2}$. Let $\mathcal{F}_{1}, \dots, \mathcal{F}_{n}$ be finite families of translates of $K$, with $n \geq 2$. Suppose that $A \cap B \neq \emptyset$ for every $A \in \mathcal{F}_{i}$ and $B \in \mathcal{F}_{j}$ with $i \neq j$.

Dol’nikov's generalized conjecture. There exists $j \in \{1,2,\dots,n\}$ such that $\bigcup_{i \neq j} \mathcal{F}_{i}$ can be pierced by $3$ points.

The statement strengthens the three-family conjecture and is motivated by colorful analogues of classical piercing theorems. The source records related results for circles, but leaves this general translate version as a conjecture.