Erdős's jump conjecture for hypergraph Turán densities

For an integer $r\ge 2$, an $r$-graph is an $r$-uniform hypergraph. A real number $\alpha\in[0,1]$ is a jump for $r$ if there exists a constant $c>0$ such that, for every $\epsilon>0$ and every integer $m\ge r$, there is an integer $n_0(\epsilon,m)$ such that every $r$-graph with at least $n_0(\epsilon,m)$ vertices and density at least $\alpha+\epsilon$ contains a subgraph with $m$ vertices and density at least $\alpha+c$. Erdős's jump conjecture. Every $\alpha\in[0,1)$ is a jump for every $r\ge 2$. Erdős proved that every $\alpha\in[0,r!/r^r)$ is a jump for $r\ge3$, while the conjecture concerns the full interval $[0,1)$ and its status is not established by the supplied text.