Piercing families of convex sets in the plane that avoid a certain subfamily with lines

We define a $C(k)$ to be a family of $k$ sets $F_1,\dots,F_k$ such that $\conv(F_i\cup F_{i+1})\cap \conv(F_j\cup F_{j+1})=\emptyset$ when $\{i,i+1\}\cap \{j,j+1\}=\emptyset$ (indices are taken modulo $k$). One can visualize the union of the convex hulls $\conv(F_j\cup F_{j+1})$ as a closed loop in the plane with no crossings. We show that if $\F$ is a family of compact, convex sets that does not contain a $C(k)$, then there are $k-2$ lines that pierce $\F$. Additionally, we give an example of a family of compact, convex sets that contains no $C(k)$ and cannot be pierced by $\left\lceil \frac{k}{2} \right\rceil -1$ lines.

This is related to and continues a line of work introduced by Eckhoff who asked how many lines it takes to pierce families of convex sets in the plane for which every $k$ sets in the family can be pierced by some line.