Linear Tur'an Numbers of Four-Edge Uniform Paths via Incidence Rank

Let $P_4^r$ denote the $r$-uniform expansion of the graph path with four edges. We prove that every $n$-vertex linear $r$-uniform $P_4^r$-free hypergraph has at most $(r+1)n/r$ edges, resolving a conjecture of Adak and Verma for every $r \geq 2$. Equality holds precisely for vertex-disjoint unions o