Conjecture on decompositions of complete uniform hypergraphs into regular Hamiltonian cycles

Let $K_n^h$ denote the complete $h$-uniform hypergraph on $n$ vertices, and let $bb K_n^h$ be the hypergraph with $bb$ copies of every edge. A regular Hamiltonian cycle is a Hamiltonian cycle that is regular as a hypergraph. Hamiltonian-cycle decomposition conjecture. For sufficiently large $n$, $\lambda K_n^h$ can be decomposed into regular Hamiltonian cycles if and only if

$$n\mid \lambda \binom{n}{h}.$$

This conjecture proposes that the divisibility condition obtained by counting edges is also sufficient for decomposing a sufficiently large complete uniform hypergraph into regular Hamiltonian cycles.