The carrying-capacity conjecture for balanced multi-patch logistic systems

Let $\alpha_1,\ldots,\alpha_n$ be the patch growth parameters, let $\Gamma$ be the migration matrix, let $K_1,\ldots,K_n$ be the patch carrying capacities, and let $X_T^\ast(\beta)$ denote the asymptotic total population under dispersal rate $\beta$. The condition

$$\left(\frac{1}{\alpha_1},\ldots,\frac{1}{\alpha_n}\right)^T\in\ker\Gamma$$

means that the vector of reciprocal growth parameters lies in the kernel of the migration matrix.

Carrying-capacity conjecture. If

$$\left(\frac{1}{\alpha_1},\ldots,\frac{1}{\alpha_n}\right)^T\in\ker\Gamma,$$

then

$$X_T^\ast(\beta)\geq\sum_{i=1}^n K_i,\qquad\text{for all }\beta\geq 0.$$

This conjecture proposes that balanced migration cannot reduce the asymptotic total population below the sum of the isolated patch carrying capacities. It extends the preceding proposition beyond the cases already identified, but no resolution is supplied here.