Spheres under a Cauchy-Riemann inequality in $E(κ,τ)$, and non-CMC Hopf tori

For a smooth, connected, oriented surface $Σ$, topologically a sphere, immersed in the homogeneous space $E(κ,τ)$ with $τ\ne0$, we show that a one-sided Cauchy--Riemann-type inequality on the original Abresch--Rosenberg differential $\mathcal{Q}_{AR}$ -- much weaker than requiring $\mathcal{Q}_{AR}$