The odd-order latin-cuboid transversal conjecture

A latin $n$-cuboid of order $q$ is an $n$-dimensional array with one dimension of length less than $q$ and the remaining dimensions of length $q$, whose lines contain different symbols. A layer is obtained by fixing one coordinate, and a transversal is a set of cells meeting every layer in at most one cell and having pairwise different symbols. Odd-order latin-cuboid transversal conjecture. Every $(2q-1)$-layer latin $n$-cuboid of odd order $q$ has transversals. The source presents this as a proposed generalization of known results for row-latin rectangles, but gives no proof or resolution status for the cuboid statement.