Linear average-length conjecture for the Zeckendorf Game

For a uniformly random Zeckendorf Game on $n$ initial copies of $1$, let $M_n$ be the number of moves until the Zeckendorf decomposition of $n$ is reached, and let $\mathbb{E}[M_n]$ denote its average length. Linear average-length conjecture. The average game length is linear in $n$, namely $\mathbb{E}[M_n]=\Theta(n)$ as $n$ tends to infinity. The conjecture proposes linear growth for the expected duration of a randomly played game; the supplied text gives no proof or resolution.