The extremal bound for hypergraphs with circumference less than the uniformity

Let $r$ be a positive integer and let $\mathcal H$ be an $r$-uniform hypergraph on $n$ vertices with no cycle of length $r$ or longer. The extremal bound conjecture. Then

$$e(\mathcal H) \leq \max\left\{\frac{r-1}{r}(n-1),\, n-r+1\right\}.$$

This conjecture proposes that the two constructions described in the paper—the relevant block-trees and the example attaining $n-r+1$ edges—give the optimal bound across the phase transition at $r=k$.