Casselgren, Markström and Pham's hypercube precoloring-extension conjecture

Let $K_{n,n}^d$ denote the $d$-fold Cartesian product of $K_{n,n}$ with itself. A proper edge-precoloring is a coloring of a subset of the edges in which adjacent precolored edges receive different colors.

Casselgren, Markström and Pham's conjecture. If $n$ and $d$ are positive integers, and $\varphi$ is a proper edge-precoloring of $K_{n,n}^d$ with at most $nd-1$ precolored edges, then $\varphi$ extends to a proper $nd$-edge-coloring of $K_{n,n}^d$.

The paper identifies this as a conjecture of Casselgren, Markström and Pham and explains that the balanced complete-bipartite Cartesian-product conjecture would imply it. Its status remains open in the supplied text.