Matching upper-bound conjecture for the linear Turán number of the four-edge hypertree

Let $r\geq 3$ and let $P_4^r$ denote the four-edge $r$-uniform hypertree considered in the paper. For an $r$-uniform hypergraph $H$, write $ex_r^{\mathrm{lin}}(n,P_4^r)$ for the maximum number of edges in an $n$-vertex linear $r$-uniform hypergraph containing no copy of $P_4^r$.

The $P_4^r$ upper-bound conjecture.

$$ex_r^{\mathrm{lin}}(n,P_4^r) \leq \dfrac{(r+1)n}{r}.$$

Equality holds if and only if the linear $r$-uniform hypergraph is the union of disjoint Steiner systems $S(2,r,r^2)$, given that the Steiner system $S(2,r,r^2)$ exists.

The conjecture would provide a matching upper bound for the lower bound established under the assumptions that $r^2\mid n$ and that $S(2,r,r^2)$ exists. The authors note that the corresponding result is known for $r\in\{2,3\}$, while obtaining the general upper bound is substantially more difficult.