Generalized Uniformness Conjecture for Kolakoski sequences

Let $m,n>0$ with $m+n$ odd, and let $K(m,n)$ be the unique infinite sequence with values in $\{m,n\}$ and first term $m$. For an integer $k\geq 0$, define sequences $t^{(i)}$ for $i\geq 1$ recursively by

$$t^{(1)}=m^k,\qquad t^{(i+1)}=C(K(m,n)_i,t^{(i)}).$$

Here $C$ is the sequence operation used in the paper, $K(m,n)_i$ denotes the $i$th term of $K(m,n)$, and $m^k$ denotes the length-$k$ sequence consisting of $k$ copies of $m$. For $x\in\{m,n\}$ and $t\in\{m,n\}^k$, let $S$ be the subset of $\mathbb N$ consisting of all $i$ such that $K(m,n)_i=x$ and $t^{(i)}=t$.

Generalized Uniformness Conjecture. The asymptotic density of $S$ equals $2^{-k-1}$.

For $k=0$, $m=1$, and $n=2$, this reduces to the classical conjecture that the limiting density of $1$ in the Kolakoski sequence is $1/2$. The paper presents a partial result, but the general conjecture remains open.