On the orthogonal Gr\"unbaum partition problem in dimension three

Gr\"unbaum's equipartition problem asked if for any measure on Rᵈ there are always d hyperplanes which divide Rᵈ into 2ᵈ μ-equal parts. This problem is known to have a positive answer for d≤ 3 and a negative one for d≥ 5. A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for d≤ 2 and there is reason to expect it to have a negative answer for d≥ 3. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of $8n$ in R³ can be split evenly by $3$ mutually orthogonal planes. To our surprise, it seems the probability that a random set of $8$ points chosen uniformly and independently in the unit cube does not admit such a partition is less than $0.001$.