Coexact 1-Laplacian spectral gap and exponential growth of a group

Let $\mathrm{\Gamma }$ be a discrete finitely presented group. Pick any system $S$ of generators in $\mathrm{\Gamma }$. In Cayley graph $\mathrm{C}\mathrm{a}\mathrm{y}(\mathrm{\Gamma })=\mathrm{C}\mathrm{a}\mathrm{y}(\mathrm{\Gamma },S)$ with edge set $E$, glue with oriented polygons all the group relations translated to all the points of $\mathrm{\Gamma }$; denote the obtained simply connected complex by ${\mathrm{C}\mathrm{a}\mathrm{y}}^{(2)}(\mathrm{\Gamma })$. We study non-negative Hodge–Laplace operator ${\mathrm{\Delta }}_1$ on edge functions which is defined via complex ${\mathrm{C}\mathrm{a}\mathrm{y}}^{(2)}(\mathrm{\Gamma })$; ${\mathrm{\Delta }}_1$ acts on \ell^2_{0,c}(E):= \mathrm{clos}_{\ell^2(E)} \left\{finitely supported closed $1$-(co)chains in \mathrm{Cay}^{}(Γ)\right\}. We prove the following implication in the spirit of Kesten Theorem: if ${\mathrm{\Delta }}_1{\mathrm{\mid }}_{{\mathrm{\ell }}_{0,c}^2(E)}$ has a spectral gap then $\mathrm{\Gamma }$ either has exponential growth or is virtually $\mathbb{Z}$.