Central limit conjecture for degree growth of tame automorphisms

Let $Q$ be the affine quadric threefold and let $G$ be its tame automorphism group. Let $\mu$ be a symmetric atomic measure on $G$, let $\nu_n=\mu^{*n}$ be its $n$-fold convolution, and let $g_n$ be distributed according to $\nu_n$. Write $\lambda(\mu)$ for the degree-growth exponent. Central limit conjecture. The limit

$$\sigma^2:=\lim_{n\rightarrow +\infty}\dfrac{1}{n}\int_G(\log\operatorname{deg}(g)-\lambda(\mu)n)^2\,d\nu_n(g)$$

exists, and the random variables

$$\dfrac{\log\operatorname{deg}(g_n)-\lambda(\mu)n}{\sqrt{n}}$$

converge to the normal distribution law $\mathcal{N}(0,\sigma^2)$. This is proposed as a central limit theorem for degree growth in random walks on the tame group, analogous to central limit theorems for random products of matrices and mapping classes; its resolution is not given in the supplied text.