Upper-bound conjecture for the growth rate in the three-patch seasonal system

Assume $a>0>b$ and $a+2b<0$, and let $\Lambda(m,T)$ be the growth rate of the three-patch seasonal system under consideration, with migration rate $m$ and period $T$. Upper-bound conjecture. For all $m>0$ and $T>0$,

$$\Lambda(m,T)\leq\frac{a+b}{2}.$$

The bound would constrain the growth rate uniformly over migration intensity and season length in this explicit example. The source provides no proof or resolution of the conjecture.