Large-$p$ conjecture for Turán densities of uniform hypergraphs

Let $X$ be a finite set, and let $\mathcal{H}\subseteq\binom{X}{r}$ be an $r$-uniform hypergraph. For integers $q\geq p\geq r\geq2$, say that $\mathcal{H}$ has the $(q,p)$-property if every $q$-vertex subset contains a $p$-vertex clique. Define

$$T_r(n,q,p)=\min\{\lvert\mathcal{H}\rvert:\mathcal{H}\subseteq\binom{[n]}{r}\text{ has the }(q,p)\text{-property}\},$$

and $t_r(q,p)=\lim_{n\to\infty}T_r(n,q,p)/\binom{n}{r}$. Large-$p$ conjecture. For integers $r\geq2$ and $p$ sufficiently large,

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

The paper proves the corresponding assertion for $r=3$ when $p\geq3$, while $p=2$ corresponds to Turán's open $K_5^3$ problem. The $r=4,p=3$ case is explicitly reported to fail, so the claim is restricted to sufficiently large $p$.