Shangguan–Tamo conjecture on limits for hypergraph configurations

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no collection of $k$ edges spanning at most $s$ vertices. For positive integers $r$, $k$, and $t$, set $s=k(r-t)+t$. Shangguan–Tamo conjecture. For any positive integers $r$, $k$, and $t$, the limit

$$\pi(r,t,k):=\lim\limits_{n \to \infty} n^{-t} f^{(r)}(n;k(r-t)+t,k)$$

exists. This generalises the Brown–Erdős–Sós conjecture, corresponding to $r=3$ and $t=2$; the limit is known in several cases, including all even $k$ when $r$ is sufficiently large relative to $k$ and $t$, and $k\in\{5,7\}$ for all $r>t\ge 2$, but the general case remains open.