Glock–Kühn–Osthus high-degree conjecture for hypergraph decompositions

For integers $q>r\ge 3$, let $G$ be an $r$-uniform hypergraph on $n$ vertices. It is $K_q^r$-divisible when, for every $i\in\{0,1,\ldots,r-1\}$ and every $i$-set $S\subseteq V(G)$, the number of edges $e\in G$ containing $S$ is divisible by $\binom{q-i}{r-i}$. Let $\delta_{r-1}(G)$ denote the minimum, over all $(r-1)$-sets $S\subseteq V(G)$, of the number of edges containing $S$.

Glock–Kühn–Osthus conjecture. For each integer $r\ge 3$, there exists a constant $C>0$ such that, for all integers $q>r$ and all sufficiently large $n$, every $K_q^r$-divisible $r$-graph $G$ on $n$ vertices satisfying

$$\delta_{r-1}(G)\ge \left(1-\frac{C}{q^{r-1}}\right)n$$

admits a $K_q^r$-decomposition.

This is a high minimum-degree generalization of the Existence Conjecture for combinatorial designs, extending decomposition questions from graphs to higher-uniformity hypergraphs.