Dense hypergraphs contain large linear subhypergraphs conjecture

A 3-uniform hypergraph is linear if any two of its edges intersect in at most one vertex. Linear-subhypergraph conjecture. There is a constant $c>0$ such that for every integer $d\geq4$, if $G$ is a (not necessarily linear) 3-uniform hypergraph on $n$ vertices with

$$|E(G)|\geq n^{3-c/d},$$

then $G$ contains a linear subhypergraph with $d$ vertices and at least $d^2/100$ edges. This weaker conjecture would provide evidence toward the Latin-square Turán conjecture, but the source states that it is still unproved.