Asymptotic density of 1s in the Oldenburger–Kolakoski sequence

Determine whether the number of 1s in the length-n prefix of the Oldenburger–Kolakoski sequence is n/2 + o(n) as n tends to infinity.

Background

The Oldenburger–Kolakoski sequence is defined by the property that its nth term equals the length of its nth run. The paper notes that the sequence remains poorly understood, particularly regarding the asymptotic behavior of the number of 1s in its prefixes.

The authors identify n/2 + o(n) as a reasonable conjectured asymptotic for the number of 1s among the first n terms, but explicitly state that this has not been established. This question is distinct from the paper’s main result, which proves the corresponding density statement for Cloitre’s self-generating sequence.