On the Dales-Zelazko conjecture for Beurling algebras on discrete groups

Let $G$ be a group which is either virtually soluble or virtually free, and let $\omega$ be a weight on $G$. We prove that, if $G$ is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $\ell^1(G, \omega)$ which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Zelazko holds for these Banach algebras. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance we describe a Beurling algebra on an infinite group in which every left ideal of finite codimension is finitely generated, and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Zelazko's conjecture.