Balanced multipartite conjecture for decompositions of $k$-trees

Let $k,n\in\mathbb{N}\setminus\{1\}$. Let $T$ be a $k$-tree, meaning a $k$-uniform tree defined recursively by starting with one edge and successively adding a vertex together with an edge containing it and a $(k-1)$-set from an existing edge. Balanced multipartite $k$-tree conjecture. If $T$ has $n$ edges, then the complete balanced $k$-partite graph on $kn$ vertices admits a decomposition into copies of $T$. This is proposed as a strengthening of the bipartite tree-packing conjecture attributed in the source to Graham and Häggkvist; the paper does not report a resolution, so the assertion remains open.