The asymptotic Lyapunov exponent conjecture for the generalized Gauss map

Let $T_{r,\alpha}$ be the generalized Gauss continued-fraction map, and let $\lambda(r,\alpha)$ be the Lyapunov exponent of the orbit of $x_0\in[0,1]$ under $T_{r,\alpha}$. Let $\lVert\cdot\rVert_\infty$ denote the supremum norm. Asymptotic Lyapunov exponent conjecture. For almost every $x_0$,

$$\lim_{\alpha \rightarrow \infty}\left\lVert 2+\log r-\lambda(r,\alpha)\right\rVert_\infty=0.$$

The conjecture describes the limiting behavior of the Lyapunov exponent as the parameter $\alpha$ tends to infinity; the supplied text offers numerical evidence from the cited figures but does not indicate a proof or resolution.