Brualdi–Hollingsworth conjecture on rainbow spanning-tree decompositions

Let $K_n$ be a complete graph and let a $1$-factorization be an edge-colouring whose colour classes form a decomposition of $K_n$ into perfect matchings. A subgraph is rainbow if all its edges have distinct colours, and a collection of subgraphs decomposes $K_n$ if its members are edge-disjoint and cover every edge. Brualdi–Hollingsworth conjecture. For all $n>4$ and any $1$-factorization of $K_n$, there exists a decomposition of $K_n$ into rainbow spanning trees. The condition $n>4$ is necessary. The conjecture asks for a full decomposition, strengthening the known results guaranteeing only finitely or linearly many edge-disjoint rainbow spanning trees; an asymptotic version was known, while the paper's main theorem proves the conjecture for sufficiently large $n$.