Monotonicity conjecture for the principal Floquet exponent in dispersal-induced growth

Let $\Lambda(m,\omega)$ denote the relevant Floquet exponent for dispersal rate $m>0$ and temporal frequency $\omega>0$ in the time-periodic two-patch model, under $\bar{r}_1<0$, $\bar{r}_2<0$, and $\chi>0$. Monotonicity conjecture. The function $\Lambda(m,\omega)$ satisfies

$$\frac{\partial\Lambda}{\partial\omega}(m,\omega)<0$$

for all $m>0$ and $\omega>0$. This would imply that, for each $m\in(0,m^*)$, there is a unique transition frequency separating dispersal-induced growth from decay. The claim is motivated by numerical observations but is not proved in the source.