Higgins's perfect matching conjecture for bipartite tensor products

Let $M$ be a binary $n\times k$ matrix. Define $M\otimes M^T$ to be the symmetric $nk\times nk$ matrix indexed by pairs in $[n]\times[k]$, with entries

$$(M\otimes M^T)((x_1,y_1),(x_2,y_2))=M(x_1,y_2)M(x_2,y_1).$$

A perfect matching is an involution on the vertex set such that every vertex is joined to its image by an edge. Higgins's conjecture. If the bipartite graph with both parts of size $nk$ and bipartite adjacency matrix $M\otimes M^T$ has a perfect matching, then the ordinary graph with adjacency matrix $M\otimes M^T$ also has a perfect matching. This question arises in the study of regular finite semigroups; it remains unproven in the general case, although the paper proves special cases and reduces it to a matrix inequality.