Turán codegree conjecture for complete hypergraphs

For integers $k\geq 2$, let $K_{k+1}^k$ be the complete $k$-graph on $k+1$ vertices. For a $k$-graph $H$, let $\delta_{k-1}(H)$ be its minimum $(k-1)$-degree, and let $\pi_{k-1}(K_{k+1}^k)$ denote the limiting normalized minimum $(k-1)$-degree threshold for $K_{k+1}^k$-free $k$-graphs. The conjecture.

$$\pi_{k-1}(K_{k+1}^k)=\frac{1}{2}\quad\text{for all }k\geq 2.$$

The lower bound $\pi_{k-1}(K_{k+1}^k)\geq 1/2$ is given by the paper's construction. The conjecture asks whether this bound is sharp for every $k$; the source provides no resolution.