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 $\text{conv}(F_i\cup F_{i+1})\cap \text{conv}(F_j\cup F_{j+1})=\mathrm{\varnothing }$ when $\{i,i+1\}\cap \{j,j+1\}=\mathrm{\varnothing }$ (indices are taken modulo $k$). We show that if $\mathcal{F}$ is a family of compact, convex sets that does not contain a $C(k)$, then there are $k-2$ lines that pierce $\mathcal{F}$. 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.