Stable constant weighted mean curvature hypersurfaces in anti-Gaussian space

We prove that in the anti-Gaussian space $({\mathbb{R}}^{m+1},{\bar{g}}_{\text{Euc}},e^{\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}4}dx)$, $m\ge 2$, round spheres centered at the origin are the only closed, connected, two-sided immersed hypersurfaces with constant weighted mean curvature that are stable under weighted-volume-preserving variations.