The Maximum Criterion conjecture for linear box fragmentation processes

Let $\mathfrak{B}_0\supset \mathfrak{B}_1\supset\cdots$ be an $m$-dimensional linear-fragmentation process, and for $1\leq d\leq m$ let $\mathfrak{m}_d^{(N)}$ denote the maximum product of $d$ side lengths of $\mathfrak{B}_N$. A sequence has strong Benford behavior in base $B$ when its significands are distributed according to Benford's law in the strong sense. Maximum Criterion conjecture. Every linear fragmentation process satisfies the Maximum Criterion in all dimensions $1\leq d\leq m$: if $\mathfrak{m}_d^{(N)}$ converges to strong Benford behavior in base $B$ as $N\to\infty$, then the corresponding $d$-volume $V_d^{(N)}=\operatorname{Vol}_d(\mathfrak{B}_N)$ also converges to strong Benford behavior in base $B$. This conjecture was posed in the cited work; the present paper says it resolves the conjecture only under mild conditions, so the unrestricted assertion remains open.