= Solution
Only the <Laplacian eigenfunctions> whose eigenvalues lie in the intersection of the unstable band $(q_-,q_+)$ with the allowed spectrum can grow. A lower bound $k^2>k_0^2$ removes all unstable modes if $k_0^2\geq q_+$, even when the unrestricted threshold in (iii) is exceeded.
If all real wavenumbers above the cutoff are possible, an equivalent strict threshold is
$$
\beta\chi_0n_0>AD+\gamma+\inf_{q>k_0^2}\left(Dq+\frac{A\gamma}{q}\right).
$$
It is unchanged when $k_0^2\leq\sqrt{A\gamma/D}$, and becomes $AD+\gamma+Dk_0^2+A\gamma/k_0^2$ when $k_0^2>\sqrt{A\gamma/D}$. A bounded geometry generally gives a discrete spectrum, so the stronger requirement is that an actual allowed eigenvalue falls in the band.
$$
\boxed{\text{Instability requires an allowed }k^2\in(q_-,q_+).}
$$
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