Uniqueness of subrows from parity-history integrals

Let $u$ and $w$ be two subrows of the Kolakoski sequence, and let $P_n(u)$ denote the parity-history integral associated with $u$ at index $n$. Parity-history uniqueness conjecture. If

$$P_n(u)=P_n(w)\quad\text{for every }n\geq 0,$$

then $u=w$. This is presented as a stronger conjecture than the nonexistence of $\infty$-regular subrows: subrows with exactly the same history of parity in their integrals should coincide. The supplied context does not state whether any cases are known or resolved.