Exact upper bound for edges of edge-minimal hypertrees

Let $k$ be a positive integer, and let $\mathcal{F}=(V,\mathcal{E})$ be a $k$-uniform edge-minimal hypertree on $n=|V|$ vertices. Edge-minimal hypertree upper-bound conjecture. One has

$$|\mathcal{E}|\leq \frac{1}{k-1}\binom{n}{2}.$$

The paper presents this as the conjectured upper bound and proves only an easier $O(n^3)$ upper bound, while its construction shows that the order $n^2$ bound is asymptotically sharp.