  We continue the study of the bosonic $O(N)^3$ model with quartic interactions
and long-range propagator. The symmetry group allows for three distinct
invariant $\phi^4$ composite operators, known as tetrahedron, pillow and
double-trace. As shown in arXiv:1903.03578 and arXiv:1909.07767, the
tetrahedron operator is exactly marginal in the large-$N$ limit and for a
purely imaginary tetrahedron coupling a line of real infrared fixed points
(parametrized by the absolute value of the tetrahedron coupling) is found for
the other two couplings. These fixed points have real critical exponents and a
real spectrum of bilinear operators, satisfying unitarity constraints. This
raises the question whether at large-$N$ the model is unitary, despite the
tetrahedron coupling being imaginary. In this paper, we first rederive the
above results by a different regularization and renormalization scheme. We then
discuss the operator mixing for composite operators and we give a perturbative
proof of conformal invariance of the model at the infrared fixed points by
adapting a similar proof from the long-range Ising model. At last, we identify
the scaling operators at the fixed point and compute the two- and three-point
functions of $\phi^4$ and $\phi^2$ composite operators. The correlations have
the expected conformal behavior and the OPE coefficients are all real,
reinforcing the claim that the large-$N$ CFT is unitary.
