Pikhurko's saturation limit conjecture for hypergraphs

Let an $r$-graph be a hypergraph whose edges are $r$-element subsets of its vertex set. For an $r$-graph $F$, let an $F$-saturated $r$-graph be an $r$-graph containing no copy of $F$ such that adding any extra edge creates a copy of $F$, and let

$$\operatorname{Sat}(F,n)=\min\{e(H):|H|=n\text{ and }H\text{ is }F\text{-saturated}\}.$$

Pikhurko's conjecture. For every $r$-graph $F$, the limit

$$\lim_{n\rightarrow\infty}\frac{\operatorname{Sat}(F,n)}{n^{r-1}}$$

exists. Pikhurko established an $O(n^{r-1})$ upper bound for finite forbidden families, motivating this hypergraph generalization of Tuza's conjecture. The source gives no resolution of the asserted limit, so it remains open here.