Monotonicity, convexity and limiting value of Lyapunov exponents for the voter model

Let $\lambda_p(\kappa)$ be the $p$th Lyapunov exponent of the voter-model catalyst, where $\kappa\in[0,\infty)$ is the reactant diffusion constant, $p\in\mathbb{N}$, and $\rho\in(0,1)$ and $\gamma>0$ are model parameters.

Voter-model Lyapunov-exponent conjecture. On $[0,\infty)$, $\kappa\mapsto\lambda_p(\kappa)$ is strictly decreasing and convex, with

$$\lim_{\kappa\to\infty}\lambda_p(\kappa)=\rho\gamma.$$

The paper notes that the analogous properties are known for the independent simple random walk and simple exclusion process catalysts. Their validity for the voter model is left as a conjecture.