Weak balls conjecture for biregular bipartite multigraphs

Let $M$ be a binary $n\times k$ matrix with non-negative integer entries, row sums $k$, and column sums $n$. Let $M\otimes M^T$ be the matrix associated with the corresponding tensor-product adjacency structure. Weak balls conjecture. The $(non\text{-}bipartite)$ $nk$-regular multigraph with $nk$ vertices and adjacency matrix $M\otimes M^T$ has a perfect matching. This conjecture is equivalent to Higgins's conjecture, and the paper proves special cases; the general case remains open.