Periodicity conjecture for star-Kayles games

Let $STARKAYLES_{k,n}$ be the graph game in which the underlying graph is obtained by starting with a star graph on $k$ vertices and extending one branch to a path of edge-length $n$. A move selects a vertex and removes a positive number of edges incident to that vertex; terminal positions are edgeless graphs. Star-Kayles periodicity conjecture. For every fixed $k$, the sequence $STARKAYLES_{k,n}$ is periodic in $n$, with period a multiple of $12$. The authors report computational verification for small values of $k$ and motivate the conjecture by the expectation that a fixed star does not affect the asymptotic behaviour.