Turán's conjectures for $3$-uniform hypergraphs

Let $t_r(q,p)=\lim_{n\to\infty}T_r(n,q,p)/\binom{n}{r}$, where $T_r(n,q,p)$ is the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices having the property that every $q$-vertex set contains a $p$-vertex clique. Turán's conjectures.

$$t_3(4,3)=\frac{4}{9},\qquad t_3(5,3)=\frac{1}{4}.$$

Even though these conjectures have been around for quite a long time, neither statement was proved according to the source. They concern the asymptotic minimum edge densities forced by the $(q,p)$-property and remain open in the stated source context.