Brualdi–Busch conjecture on edge-disjoint 1-factors in realizations of degree sequences

Let $d70=(d_{1},\ldots,d_{n})$ be a non-increasing degree sequence with even $n$, and let $\mathcal{D}_{k}(\pi)=(d_{1}-k,\ldots,d_{n}-k)$. A 1-factor is a spanning $1$-regular subgraph, and a realization of $d70$ is a graph having $d70$ as its degree sequence.

Brualdi–Busch conjecture. Some realization of $d70$ has $k$ edge-disjoint $1$-factors if and only if $\mathcal{D}_{k}(\pi)$ is graphic.

This strengthens Kundu's $k$-factor theorem by requiring the $k$-factor to decompose into $k$ edge-disjoint $1$-factors. Earlier results establish the analogous conclusion for at most $\min\{4,k\}$ factors; the conjecture asks for the full value $r=k$.