Beta-universality conjecture for bounded-type Benford sequences

Let $(a_n)$ be a sequence satisfying Benford's Law through equidistribution dynamics, and suppose the continued fraction of its rotation number is of bounded type, meaning that $\sup_k a_k<\infty$. Let $\beta_{\mathrm{eff}}(N)$ denote its effective conditional-mutual-information decay exponent. Beta-universality conjecture. For every such sequence,

$$\beta_{\mathrm{eff}}(N)=2-\frac{2\log\log N}{\log N}+O\left(\frac{1}{\log N}\right),$$

with the same leading coefficient $2$. The conjecture formalizes the computationally observed quadratic decay for bounded-type bases. Establishing it would require controlling the asymptotic CMI error uniformly across all bounded-type equidistribution dynamics.