A note on Hilbert transform over lattices of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$

González-Pérez, Parcet and Xia introduced recently a framework to study $L_p$-boundedness of certain families of idempotent multipliers on von Neumann algebras. It includes symbols $m\text{\hspace{0.17em}⁣}:{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})\to \mathbb{R}$ arising from lifting the indicator function of a partition $\{{\mathrm{\Sigma }}^{+},{\mathrm{\Sigma }}^{+},{\mathrm{\Sigma }}^{-}\}$ of the hyperbolic space ${\mathbb{H}}^3$ to its isometry group ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$. The boundedness of $T_m$ on $L_p(\mathcal{L}{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C}))$ was disproved by Parcet, de la Salle and Tablate. Nevertheless, we will show that this Fourier multiplier is bounded when restricted to the arithmetic lattices ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z}[\sqrt{-n}])$, solving a question left open by the first named authors.