Density conjectures for the two special classes of maximally circlable numbers

Let $f(k)$ and $g(k)$ be the integer-valued sequences defined in the paper's first and second special classes, respectively. Call a maximally circlable number strong if it is immediately followed by a non-maximally circlable number. Special-class density conjecture. The numbers $f(k)$ are all maximally circlable, and the strong numbers among them have positive density. The numbers $g(k)$ contain infinitely many maximally circlable numbers, and the strong numbers among them have positive density. The two assertions are motivated by the apparent symmetries and the computed examples of the special circles; the paper supplies no proof or resolution.