Turán's conjecture for complete 3-uniform hypergraphs

Let $K_k^{(3)}$ be the complete $k$-vertex $3$-uniform hypergraph, and let $\pi(K_k^{(3)})$ denote its Turán density.

Turán's conjecture. For any integer $k\geqslant 4$, we have

$$\pi(K_k^{(3)})=1-\left(\frac{2}{k-1}\right)^2.$$

This is a central open problem in extremal hypergraph theory; the exact Turán density is known only for a few small values of $k$.