Relatively geometric actions of Kähler groups on CAT(0) cube complexes

We prove that for $n\ge 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ cube complex, in the sense of Einstein and Groves. As a consequence, if $\mathrm{\Gamma }$ is a non-uniform lattice in a non-compact semisimple Lie group $G$ without compact factors that admits a relatively geometric action on a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ cube complex, then $G$ is commensurable with $\text{SO}(n,1)$. We also prove that if a Kähler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ cube complex, then it is virtually a surface group.