Density-half conjecture for discrete stick fragmentation

Let $\mathfrak{S}\subseteq\mathbb{Z}_+$ be a stopping set such that the natural density exists, and write

$$r=\lim_{n\to\infty}\frac{|[1,n]\cap\mathfrak{S}|}{n}.$$

Assume that $r>0$. Density-half conjecture. The set of dead stick lengths approaches Benford behavior if and only if $r=1/2$. This conjecture proposes that the density of the stopping set is the only essential feature governing Benfordness in the discrete fragmentation process; the precise scope of the assertion remains open.