Kalai's tight-tree conjecture for tight paths

For $n \geq r \geq 2$ and $k \geq 1$, let $P_k^r$ be the tight $k$-path in an $r$-uniform hypergraph, and let ${\rm ex}(n,P_k^r)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no tight $k$-path. Kalai's conjecture.

$${\rm ex}(n,P_k^r) \leq \frac{k-1}{r}{n \choose r-1}.$$

This is presented as a special case of Kalai's conjecture on tight trees, generalizing the Erdős–Sós conjecture. The claim concerns the extremal number of tight paths in uniform hypergraphs; the source does not state whether it has been resolved.