Nonexistence of infinity-regular subrows in the Kolakoski sequence

Let $S$ be the Kolakoski sequence and let $\mathcal{SR}(S)$ denote its set of subrows. An $\infty$-regular subrow is a subrow that is $k$-regular for every $k$. Nonexistence conjecture. There are no $\infty$-regular subrows in $\mathcal{SR}(S)$. This conjecture would imply the stated consequence in Theorem 1; it arises from the apparent increase in the number of independent conditions required for a finite word to be $k$-regular as $k$ grows.