The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence:

$$OK = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,\ldots).$$

For $n \geq 1$, let $a(n)$ be the number of indices $h \leq n$ such that $OK(h) =1$, and let $b(n)$ be the number of $h \leq n$ such that $OK(h) = 2.$

QUESTION. Is $a(n) = b(n)$ for infinitely many $n$?

For example, the first ten runs are $1,22,11,2,1,22,1,22,11,2$, and the lengths of these runs are $1,2,2,1,1,2,1,2,2,1$.

In the OEIS, the sequence is A000002. There are several related easily stated unsolved problems, such as the conjecture that the limiting density of $1$'s is $1/2$.