Hypergraph tree-packing conjecture

Let $k,n\in\mathbb{N}\setminus\{1\}$. Let $\mathcal{T}$ be a family of $k$-trees, where a $k$-tree is the recursively defined $k$-uniform tree obtained from one edge by repeatedly adding a vertex and an edge containing it and a $(k-1)$-set from an existing edge. Suppose that, for every $i\in[n-k+1]$, the family $\mathcal{T}$ contains $\binom{n-i-1}{k-2}$ trees with $i$ edges. Hypergraph tree-packing conjecture. Then $K_n^{(k)}$ admits a decomposition into $\mathcal{T}$. This is a proposed $k$-graph analogue of the tree-packing conjecture; the source gives no resolution, and the exact decomposition statement remains open.