Conjecture on Benford behavior of lower-dimensional volumes

Let an $m$-dimensional object undergo a fragmentation process, and let $d$-dimensional volumes with $d<m$ be the corresponding lower-dimensional geometric measures of its pieces. The Benford distribution is the distribution of significands specified by Benford's law.

Lower-dimensional volume conjecture. A $d$-dimensional volume for an $m$-dimensional object, where $d<m$, undergoing a fragmentation process follows the Benford distribution for the appropriate set of conditions.

This generalizes the rectangle-perimeter conjecture to arbitrary lower-dimensional volumes. The phrase “appropriate set of conditions” is not specified in the supplied text, and no resolution is given.