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

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