Hypergraph Nash-Williams conjecture for clique decompositions

Let $q>r\geq 3$ be integers. An $r$-graph is an $r$-uniform hypergraph, and it is $K_q^r$-divisible when it satisfies the divisibility conditions necessary for a decomposition into copies of the complete $r$-graph $K_q^r$. Write $\delta(G)$ for the minimum $(r-1)$-degree and $v(G)$ for the number of vertices. Hypergraph Nash-Williams conjecture. If $G$ is a sufficiently large $K_q^r$-divisible $r$-graph such that

$$\delta(G)\geq\left(1-\Theta_r\left(\frac{1}{q^{r-1}}\right)\right)\cdot v(G),$$

then $G$ admits a $K_q^r$-decomposition. This conjecture generalizes Nash-Williams' triangle-decomposition conjecture to uniform hypergraphs. The source states that the paper proves a bound of order $1-c/q^{r-1+o(1)}$, approximately confirming the conjectured dependence on $q$, but does not establish the conjecture in its full stated form.