Refined covering-number conjecture for intersecting segment hypergraphs

Let $H$ be an intersecting $r$-segment hypergraph, meaning that every two edges of $H$ intersect, and let $r\geq 5$. The covering number $\tau(H)$ is the minimum size of a vertex set meeting every edge.

Refined intersecting-segment conjecture. If $H$ is an intersecting $r$-segment hypergraph and $r\geq 5$, then

$$\tau(H)\leq \left\lceil\frac{r}{2}\right\rceil.$$

This is presented as a refinement of the main covering bound for segment hypergraphs and is motivated by analogous bounds related to the generalized Tuza conjecture for certain linear hypergraphs. The source does not establish the conjecture in general.