Benford behavior for additive decompositions stopped at evens or primes

Consider an additive decomposition process starting from a stick of integer length $L$: choose uniformly an integer cut point $c\in[1,L-1]$, replace the stick by pieces of lengths $c$ and $L-c$, and continue recursively. A piece stops decomposing when its length is $1$ or belongs to a prescribed sequence $\{a_n\}$. Benford behavior conjecture. The resulting stick lengths follow Benford's Law for many choices of $\{a_n\}$; in particular, this holds when either

$$\{a_n\}=\{2n\}$$

or $\{a_n\}$ is the set of all prime numbers. This proposes Benford behavior for discrete additive decomposition processes stopped at even lengths or prime lengths, supported in the paper by numerical evidence and by analogy with the continuous model.