Frankl–Huang–Rödl conjecture on local Turán densities

For integers $r\ge 2$ and sufficiently large integers $p$, let $t_r(2p+1,p+1)$ denote the local Turán density of the complete $r$-uniform hypergraph on $p+1$ vertices in a hypergraph on $2p+1$ vertices. Frankl–Huang–Rödl conjecture.

$$t_r(2p+1,p+1)=\frac{1}{2^{r-1}}.$$

Frankl, Huang and Rödl proposed this as an extension of their earlier result for $3$-uniform hypergraphs. The paper describes local Turán density problems as widely open, and no resolution of this assertion is supplied.