Asymptotic persistence density conjecture for Benford sequences

Let $b$ range over integer bases greater than $1$. A base is B-EQUI-PERS at sample size $N$ when there exists a continued-fraction convergent denominator $q_k\leq N$ such that $a_{k+1}q_k>N\log N$, where $a_{k+1}$ is the next partial quotient of the continued fraction of $\log_{10}(b)$. Asymptotic persistence density conjecture. The persistence density among integer bases converges:

$$\rho_\infty=\lim_{B\to\infty}\frac{\#\{b\leq B:b\text{ is B-EQUI-PERS}\}}{B-1}=\frac{1}{12}.$$

Extended computation gives densities close to $1/12$, while a Gauss--Kuzmin heuristic accounts for only part of the observed rate. A proof would require understanding the joint distribution of continued-fraction partial quotients and denominators for integer logarithms, including correlations not captured by existing theory.