  We show that an integrable four-dimensional non-unitary field theory that was
recently proposed as a certain limit of the $\gamma_i$-deformed $\mathcal{N}=4$
SYM theory is incomplete and not conformal -- not even in the planar limit. We
complete this theory by double-trace couplings and find conformal one-loop
fix-points when admitting respective complex coupling constants. These
couplings must not be neglected in the planar limit, as they can contribute to
planar multi-point functions. Based on our results for certain two-loop planar
anomalous dimensions, we propose tests of integrability.
