The Kolakoski sequence digit distribution conjecture

Let $K=(K_n)_{n=1}^{\infty}$ be the classical Kolakoski sequence on the alphabet $\{\mathtt{1},\mathtt{2}\}$, and let $o_n=|\{i:K_i=\mathtt{1},\ 1\leq i\leq n\}|$ be the number of occurrences of $\mathtt{1}$ among the first $n$ terms. Kolakoski digit distribution conjecture. The limit

$$\lim_{n\to\infty}\frac{o_n}{n}$$

exists and equals $\frac{1}{2}$. This is a basic unresolved question about the digit distribution of the Kolakoski sequence; both the existence of the limit and its value remain open.