Strict monotonicity conjecture for the growth rate under time-independent migration

Let $\Lambda(m,T)$ denote the growth rate for migration intensity $m$ and period parameter $T$, and suppose that the migration matrix is time independent. Let $r_i(\tau)$ denote the local growth rates in patch $i$ at time $\tau$. Strict monotonicity conjecture. The function

$$T\mapsto\Lambda(m,T)$$

is strictly increasing for all $m$, except when all $r_i(\tau)$ are equal. The conjecture concerns the non-symmetric time-independent migration case; strict monotonicity is known for time-independent symmetric migration, while it fails for time-dependent migration.