Glock, Kühn and Osthus's cycle-decomposition threshold conjecture

Let $k$ be an integer with $k\geq 3$. For a $k$-graph $H$, a cycle-decomposition is an edge partition of $H$ into tight cycles, and the cycle-decomposition threshold $\delta^{(k)}_{\mathrm{cycle}}$ is the least $d>0$ such that, for every $\varepsilon>0$, all sufficiently large $k$-graphs $H$ with minimum codegree at least $(d+\varepsilon)n$ and every vertex degree divisible by $k$ admit a cycle-decomposition. Glock, Kühn and Osthus's conjecture.

$$\delta^{(k)}_{\mathrm{cycle}}\leq\frac{k-1}{k}.$$

The threshold is $0$ for graphs and equals $2/3$ for $3$-graphs; the conjecture asks for the general upper bound suggested by these results. Its resolution is not specified in the source.