Failure of Benford behavior for polynomial, exponential, and Fibonacci stopping sequences

Fix a monotonically increasing sequence $\{a_n\}$. In the additive decomposition process, a stick whose length belongs to $\{a_n\}$ does not decompose further; sticks of length $1$ also stop decomposing. Non-Benford decomposition conjecture. The process is not Benford if $\{a_n\}$ is any of

$$\{n^2\},\qquad \{2^n\},\qquad \{F_n\},$$

where $F_n$ is the $n$th Fibonacci number. Together with the positive examples for even and prime stopping sequences, this identifies classes of stopping sequences expected to produce or prevent Benford behavior; the supplied text does not state a proof or resolution.