Let $G$ be a discrete, finitely generated group. Let $f\in \mathbb{C} G$ be given. Consider $g\in G\setminus \operatorname{supp} f$ and let $\delta_g$ denote the Dirac delta at $g$.

Is it true that $\Vert f\Vert\le \Vert f+\delta_g\Vert$?

The norms here are in $B(\ell_2(G))$, as convolution operators on $\ell_2(G)$.