  Gr\"unbaum's equipartition problem asked if for any measure on $\mathbb{R}^d$
there are always $d$ hyperplanes which divide $\mathbb{R}^d$ into $2^d$
$\mu$-equal parts. This problem is known to have a positive answer for $d\le 3$
and a negative one for $d\ge 5$. A variant of this question is to require the
hyperplanes to be mutually orthogonal. This variant is known to have a positive
answer for $d\le 2$ and there is reason to expect it to have a negative answer
for $d\ge 3$. In this note we exhibit measures that prove this. Additionally,
we describe an algorithm that checks if a set of $8n$ in $\mathbb{R}^3$ 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 $0.001$.
