Embedding arbitrary edge-colorings of hypergraphs into regular colorings

For $\mathbf{\text{r}}=(r_1,\dots ,r_k)$, an $\mathbf{\text{r}}$-factorization of the complete $\lambda$-fold $h$-uniform $n$-vertex hypergraph $\lambda K_n^h$ is a partition of the edges of $\lambda K_n^h$ into $F_1,\dots ,F_k$ such that $F_j$ is $r_j$-regular and spanning for $1\le j\le k$. This paper shows that for $n>\frac{m-1}{1-2^{\frac{1}{1-h}}}+h-1$, a partial $\mathbf{\text{r}}$-factorization of $\lambda K_m^h$ can be extended to an $\mathbf{\text{r}}$-factorization of $\lambda K_n^h$ if and only if the obvious necessary conditions are satisfied.