Periodicity conjecture for $KAYLES_n\boxplus(*1)$

Let $KAYLES_n$ denote the Kayles graph game on a path of edge-length $n$, and let $\boxplus$ denote the selective compound operation used in the paper; write $*1$ for the one-point Nim game. Kayles selective-compound periodicity conjecture. The sequence $KAYLES_n\boxplus(*1)$ is periodic in $n$. This is presented as a related direction beyond the periodicity results currently obtained by the threshold theorem and computational verification.