The jump threshold conjecture for 3-uniform hypergraphs

For a $3$-uniform hypergraph, a number $\theta$ is called a jump if there exists $\beta<\theta$ such that every sufficiently large $3$-uniform hypergraph with edge density at least $\theta$ contains a subhypergraph of positive density whose edge density is at least $\theta$; otherwise, $\theta$ is a non-jump.

Jump threshold conjecture. All numbers in $[0,4/9)$ are jumps and all numbers in $[4/9,1)$ are non-jumps.

This conjecture would determine the complete jump spectrum for $3$-uniform hypergraphs. In particular, together with the theorem that $4/9$ is a non-jump, it would imply that $4/9$ is the exact threshold between jumps and non-jumps and would resolve Erdős's question about whether $2/9$ is a jump.