The conjectured value of the line-piercing number $L(k)$

Let $\mathcal{F}$ be a family of compact, convex sets in the plane, and let $L(k)$ denote the minimum number of lines guaranteed to pierce $\mathcal{F}$ when $\mathcal{F}$ contains no $C(k)$, where $C(k)$ is a family of $k$ sets $F_1,\dots,F_k$ such that

$$\operatorname{conv}(F_i\cup F_{i+1})\cap \operatorname{conv}(F_j\cup F_{j+1})=\emptyset$$

whenever $\{i,i+1\}\cap\{j,j+1\}=\emptyset$, with indices taken modulo $k$. The conjecture for $L(k)$. We have

$$L(k)=\left\lceil \frac{k}{2} \right\rceil.$$

The paper explains that this is the conjectured correct value because it matches the lower bound proved in the main theorem; the source does not specify whether the conjecture has been resolved.