Minimum-degree Baranyai factorization conjecture

Fix $k$ and $\varepsilon>0$. Let $G$ be an $n$-vertex $k$-uniform hypergraph, and let $\delta(G)$ denote its minimum vertex degree. A perfect matching is a set of pairwise disjoint edges covering all vertices; a decomposition into perfect matchings is a $1$-factorization.

Minimum-degree Baranyai conjecture. For all sufficiently large $n$, an $n$-vertex $k$-graph $G$ with

$$\delta(G)\ge (1/2+\varepsilon)n$$

can be decomposed into perfect matchings if and only if $k\mid n$ and $G$ is vertex-regular.

This proposes a dense minimum-degree extension of Baranyai's theorem. The divisibility and regularity conditions are necessary, and the source presents sufficiency as open.