Factorization conjecture for complete multipartite 3-uniform hypergraphs with unequal part sizes

Let $\mathscr K_{m_1,\ldots,m_n}^{3}$ be the complete multipartite 3-uniform hypergraph whose parts have sizes $m_1,\ldots,m_n$, and let $\lambda \mathscr K_{m_1,\ldots,m_n}^{3}$ denote the hypergraph with each edge having multiplicity $\lambda$. Let an $(r_1,\ldots,r_k)$-factorization partition the edge set into spanning sub-hypergraphs in which every vertex has degree $r_i$ in the $i$th factor. Factorization conjecture. The hypergraph $\lambda \mathscr K_{m_1,\ldots,m_n}^{3}$ is $(r_1,\ldots,r_k)$-factorable if and only if all parts have a common size $m$, each $r_i mn$ is divisible by $3$, and

$$\sum_{i=1}^{k} r_i=3\lambda(n-1)\binom{m}{2}.$$

This is presented as a reasonable conjecture for the generalization allowing different part sizes; the source provides no resolution, so its status remains open.