Only constant mean curvature surfaces in $E(κ,τ)$, $τ\ne0$, have a holomorphic original Abresch-Rosenberg differential

For immersed surfaces in the homogeneous spaces $E(κ,τ)$ with $τ\ne0$, we prove that the original Abresch--Rosenberg differential $[2(H+iτ)p-(κ-4τ^2)A^2]\,dζ^2$ is holomorphic if and only if the mean curvature $H$ is constant; here $p\,dζ^2$ is the Hopf differential and $A=\langleξ,\partial_ζ\rangle