Strict monotonicity conjecture for growth rates under random environmental switching

Let $L$ be an irreducible dispersal matrix, let $A_i=R_i+L$ be the matrices in the differential inclusion model, and let $x_i$ be eigenvectors of $A_i$. Assume that the eigenvectors $x_i$ are not all equal. Let $\Lambda_M(\omega)$ denote the population growth rate under random environmental switching with frequency $\omega$. Strict monotonicity conjecture. If $L$ is symmetric or $d=2$, then $\omega\mapsto \Lambda_M(\omega)$ is strictly decreasing. This conjecture extends the established monotonicity results for sufficiently slow switching in the symmetric-dispersal or two-patch cases to all switching frequencies; the paper does not establish strict decrease in general.