The multipartite semiregular factorisation conjecture for complete bipartite graphs

Let $K_{m,n}$ be the complete bipartite graph with parts of sizes $m$ and $n$, and let $R(m,n;\lambda_0,\ldots,\lambda_k)$ count partitions of its edges into spanning semiregular subgraphs with densities $\lambda_0,\ldots,\lambda_k$. For positive numbers $\lambda_0,\ldots,\lambda_k$ satisfying

$$\sum_{i=0}^k\lambda_i=1,$$

write $\binom{a}{a_0,\ldots,a_k}$ for a multinomial coefficient and define

$$R'(m,n;\lambda_0,\ldots,\lambda_k)=\frac{\displaystyle\binom{n}{\lambda_0 n,\ldots,\lambda_k n}^{\!m}\binom{m}{\lambda_0 m,\ldots,\lambda_k m}^{\!n}}{\displaystyle\binom{mn}{\lambda_0 mn,\ldots,\lambda_k mn}}(1-1/m)^{k(m-1)/2}.$$

The multipartite semiregular factorisation conjecture. If $n\to\infty$ with $2\le m\le n$ and $1\le k=o(m)$, then

$$R(m,n;\lambda_0,\ldots,\lambda_k)\sim R'(m,n;\lambda_0,\ldots,\lambda_k).$$

This generalises the known asymptotic formula for decompositions into two spanning semiregular subgraphs. The paper proves the conjecture in several regimes, including fixed $m$, sparse cases, nearly equal densities, and the Latin-rectangle case, but leaves the full range $1\le k=o(m)$ open.