Linear-growth conjecture for Split Smallest and Combine Smallest games

A Zeckendorf game starts from the Zeckendorf decomposition of $n$ and proceeds by legal combining and splitting moves; the Split Smallest and Combine Smallest games are deterministic strategies, and the number of moves is measured as the total number of moves in the resulting game.

Linear-growth conjecture. For Split Smallest, the number of moves grows linearly with $n$, with the asymptotic constant appearing numerically to be the square of the golden mean. For Combine Smallest games, the number of moves grows linearly with $n$, with the constant appearing numerically to be approximately $1.206$.

The paper presents these claims as data-supported conjectures for the two deterministic games whose behavior was not rigorously determined there.