Partition problems in high dimensional boxes

Alon, Bohman, Holzman and Kleitman proved that any partition of a d-dimensional discrete box into proper sub-boxes must consist of at least 2 ^d sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a d-dimensional box of odd size, and they asked whether the trivial construction consisting of 3 ^d boxes is best possible. We show that approximately 2.93 ^d boxes are enough, and consider some natural generalisations.