Asymptotic growth conjecture for the season-swapped three-patch system

Assume $a>0>b$ and $a+2b<0$. For the seasonal system denoted by (47ab1), let $\Lambda(m,T)$ be its growth rate, with migration rate $m$ and period $T$. Asymptotic growth conjecture. For every $T>0$,

$$\Lambda(\infty,T)=b,$$

and for every $m\geq a-b$,

$$\Lambda(m,\infty)=b.$$

These are conjectured formulas for the infinite-migration and infinite-period limits of the season-swapped system. The source does not establish either asserted limit in the indicated parameter range.