Cloitre's density conjecture for the self-generating sequence

Let $(a_n)_{n \geq 1}$ be the unique sequence over the alphabet $\\{1,2 \\}$, beginning with $a_1=1$, whose runs have sums equal to twice the corresponding sequence terms:

$$(a_n)_{n \geq 1}=\underbrace{1,1,}_2 \, \underbrace{2,}_2 \, \underbrace{1,1,1,1,}_4 \, \underbrace{2,}_2 \, \underbrace{1,1,}_2 \, \underbrace{2,}_2 \, \underbrace{1,1,}_2 \, \underbrace{2,2,}_4 \, \underbrace{1,1,}_2 \, \underbrace{2,}_2 \, \underbrace{1,1,1,1,}_4 \ldots$$

Cloitre's density conjecture. The number of $1$'s appearing in the prefix $a_1a_2\cdots a_n$ is $2n/3+o(n)$.

The paper proves the stronger bound $0\leq 3g_n-2n\leq 4$, where $g_n$ counts the $1$'s in the prefix, so the conjecture is resolved.