Large-shear conjecture for positivity of the top Lyapunov exponent

Consider the random dynamical system induced by the stochastic differential equation in the paper, and fix $a>0$ and $\beta\in\mathbb{R}$. Let $\lambda_{\rm top}$ denote its top Lyapunov exponent. Large-shear conjecture. There exists a function

$$C:\mathbb{R}\times\mathbb{R}^+\to\mathbb{R}^+$$

such that, whenever

$$b\geq C(\alpha,\sigma),$$

the top Lyapunov exponent satisfies $\lambda_{\rm top}>0$. Numerical evidence suggests that large shear leads to a positive top Lyapunov exponent, but the paper does not provide an analytical proof.