  This note provides new criteria on a unimodular group $G$ and a discrete
series representation $(\pi, \mathcal{H}_{\pi})$ of formal degree $d_{\pi} > 0$
under which any lattice $\Gamma \leq G$ with $\text{vol}(G/\Gamma) d_{\pi} \leq
1$ (resp. $\text{vol}(G/\Gamma) d_{\pi} \geq 1$) admits $g \in
\mathcal{H}_{\pi}$ such that $\pi(\Gamma) g$ is a frame (resp. Riesz sequence).
The results apply to all projective discrete series of exponential Lie groups.
