On Non-Uniform Baranyai Factorizations

For a fixed nonempty finite set $L$ of positive integers and $k=\max L\ge2$, let $\binom{[n]}{L}=\bigcup_{\ell\in L}\binom{[n]}{\ell}$ denote the non-uniform hypergraph whose edge set consists of all $\ell$-subsets of $[n]$ for $\ell\in L$. A $1$-factor of a hypergraph $H$ is a collection of pairwis