Baird–Epstein–Flint–Miller Gaussianity conjecture for random Zeckendorf games

Let $N$ be a positive integer, and consider a random Zeckendorf game on input $N$, in which an available move is chosen uniformly at random at each turn. Let the game length be the number of moves played.

Baird–Epstein–Flint–Miller Gaussianity conjecture. As $N\to\infty$, the distribution of the game length converges to a Gaussian distribution, with expectation and variance approximately $0.215N$.

This conjecture concerns the asymptotic distribution of random Zeckendorf game lengths and is motivated by numerical evidence. The source describes it as the only conjecture from the paper in which it was introduced that remained unresolved; further numerical experiments support the claim.