Győri et al.'s linear Turán conjecture for linear paths in 3-graphs

Let $G$ be an $n$-vertex linear 3-graph, meaning that any two distinct hyperedges of $G$ intersect in at most one vertex. A linear path of length $k$ is the unique acyclic Berge path of length $k$, and therefore has $2k+1$ vertices.

Győri et al.'s linear Turán conjecture. If $G$ contains no linear path of length $k\ge 5$, then the number of edges in $G$ is at most

$$\frac{k}{3}n+cn,$$

for some universal constant $c$.

This conjecture asks for the linear Turán number of linear paths of length at least five, extending the known upper bound for linear 3-graphs without Berge paths. Its status is not established by the supplied text.