Brualdi's conjecture for near transversals in latin hypercubes

A latin hypercube of dimension $d$ and order $n$ is a $d$-dimensional array filled with $n$ symbols so that every line contains all symbols exactly once. A near transversal is a collection of entries meeting all but one hyperplane in each coordinate direction exactly once, with distinct symbols. Brualdi's conjecture. Every latin hypercube has a near transversal. The source presents this as the proposed generalization of Brualdi's conjecture to latin hypercubes and states that there is no evidence against it; its resolution status is otherwise not specified.