Hypergraph Ringel conjecture for $k$-trees

Let $k,n\in\mathbb{N}\setminus\{1\}$. A $k$-tree is defined recursively: a single edge is a $k$-tree, and a $k$-tree with $\ell$ edges is obtained from one with $\ell-1$ edges by adding a vertex and an edge containing that vertex and a $(k-1)$-set contained in an existing edge. Hypergraph Ringel conjecture. If $T$ is a $k$-tree with $n$ edges, then $K_{kn+k-1}^{(k)}$ admits a decomposition into copies of $T$. This is the proposed $k$-uniform generalisation of Ringel's conjecture; the source notes that the bound is necessary for arbitrary $k$-trees and that the star case is elementary, but the general assertion remains open.