Alon–Shapira conjecture for generalized hypergraph Turán numbers

Let $f_r(n,v,e)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph such that every set of $v$ vertices spans fewer than $e$ edges. For fixed integers $2\leq k<r$ and $e\geq 4$, consider $f_r(n,e(r-k)+k+1,e)$. Alon–Shapira conjecture. For any fixed $2\leq k<r$ and $e\geq 4$, we have

$$n^{k-o(1)}<f_r(n,e(r-k)+k+1,e)=o(n^k).$$

This conjecture extends the asymptotic result of Alon and Shapira for $e=3$ to all $e\geq 4$, predicting the same intermediate growth between $n^{k-o(1)}$ and $o(n^k)$.