Asymptotic lower bound for edges of 3-uniform edge-maximal hypertrees

Let $n$ be the number of vertices, and let a 3-uniform edge-maximal hypertree be a 3-uniform hypertree to which no further edge can be added without destroying the hypertree property. Edge-maximal hypertree lower-bound conjecture. Every 3-uniform edge-maximal hypertree on $n$ vertices has at least

$$\frac{1}{2}\binom{n}{2}-O(n)$$

edges. The paper constructs 3-uniform edge-maximal hypertrees with $\frac{1}{2}\binom{n}{2}-\frac14n$ edges for even $n>2$, supporting the conjectured asymptotic lower bound; the universal lower bound is not proved.