The digit-frequency conjecture for the rational-base sequence

Let $\operatorname{sat}(k)$ be the sequence defined in the paper, and let $\langle\operatorname{sat}(k)\rangle$ denote its associated word over the alphabet $\{1,2\}$. The digit-frequency conjecture. The proportion of digits “1” in the word $\langle\operatorname{sat}(k)\rangle$ tends to $1/2$ as $k$ tends to infinity. This conjecture is motivated by numerical computations for the sequence and is presented as an open question related to the Collatz problem.