Füredi's extremal conjecture for generalized 4-cycle-free uniform hypergraphs

Let $f_k(n)$ denote the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no generalized $4$-cycle. For $k\geq 4$ and $n\in\mathbb{N}$, Füredi's conjecture.

$$f_k(n)=\binom{n-1}{k-1}+\left\lfloor\frac{n-1}{k}\right\rfloor.$$

The conjecture predicts the exact extremal edge count for this generalized $4$-cycle problem; the paper's abstract states that the authors verify the corresponding conjecture of Mubayi and Verstraëte for odd-uniform hypergraphs.