Gaussian limit conjecture for random Bergman game lengths

Let ${}_0(n)$ denote the initial state of the Bergman Game, and let $\mathcal{L}_n$ be the distribution of the length of a random game played from ${}_0(n)$, where at each state an available move is chosen uniformly at random. Gaussian limit conjecture. The distribution $\mathcal{L}_n$ converges to a Gaussian with linear mean

$$\mu_n \approx 2n+O(1).$$

This conjecture is based on numerical data for the Bergman Game; the corresponding behavior for generalized Bergman Games is expected to be similar, but no concrete results are proved in the source.