The sharp linear Turán bound conjecture for 4-edge linear paths

Let $P_4^r$ be the $r$-uniform linear path with four edges, and let $ex_r^{\mathrm{lin}}(n,P_4^r)$ denote the maximum number of edges in an $n$-vertex linear $r$-uniform hypergraph containing no copy of $P_4^r$. A Steiner system $S(2,r,r^2)$ is a design on $r^2$ vertices in which every pair of vertices lies in exactly one $r$-element block. The sharp 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.

This conjecture would make the lower bound obtained from disjoint copies of $S(2,r,r^2)$ sharp and would characterize all equality cases when the relevant Steiner system exists. The supplied text does not state whether the conjecture has been resolved.