An improved hypergraph Mantel's Theorem

In a recent paper, Chao and Yu used an entropy method to show that the Tur\'an density of a certain family $\mathcal{F}$ of $\lfloor r/2\rfloor$ triangle-like $r$-uniform hypergraphs is $r!/r^r$. Later, Liu determined for large $n$ the exact Tur\'an number $\text{ex}(n,\mathcal{F})$ of this family, and showed that the unique extremal graph is the balanced complete $r$-partite $r$-uniform hypergraph. These two results together can be viewed as a hypergraph version of Mantel's Theorem. In this paper, building on their methods, we improve both of these results by showing that they still hold with a subfamily $\mathcal{F}'\subset\mathcal{F}$ of size $\lceil r/e\rceil$ in place of $\mathcal{F}$.