Győri–Katona–Lemons conjecture for Berge paths in uniform hypergraphs

Let $r$ and $k$ satisfy $k=r+1>2$, and let $\mathcal{H}$ be an $r$-uniform hypergraph, meaning that every hyperedge has size $r$. A Berge path of length $k$ consists of $k$ distinct hyperedges $e_1,\dots,e_k$ and $k+1$ distinct vertices $v_1,\dots,v_{k+1}$ such that $v_i,v_{i+1}\in e_i$ for each $1\leq i\leq k$. Write $e(\mathcal{H})=|E(\mathcal{H})|$ and $n=n(\mathcal{H})=|V(\mathcal{H})|$. Győri–Katona–Lemons conjecture. If $\mathcal{H}$ contains no Berge path of length $k$, then

$$e(\mathcal{H})\leq \frac{n}{k}\binom{k}{r}=n.$$

This conjecture is the remaining case of the Győri–Katona–Lemons extension of the Erdős–Gallai theorem for uniform hypergraphs; the paper settles it by proving that an $r$-uniform hypergraph with more than $n$ edges contains a Berge path of length $r+1$.