  A tensorial representation of $\phi^4$ field theory introduced in Phys. Rev.
D. 93, 085005 (2016) is studied close to six dimensions, with an eye towards a
possible realization of an interacting conformal field theory in five
dimensions. We employ the two-loop $\epsilon$-expansion, two-loop
fixed-dimension renormalization group, and non-perturbative functional
renormalization group. An interacting, real, infrared-stable fixed point is
found near six dimensions, and the corresponding anomalous dimensions are
computed to the second order in small parameter $\epsilon=6-d$. Two-loop
epsilon-expansion indicates, however, that the second-order corrections may
destabilize the fixed point at some critical $\epsilon_c <1$. A more detailed
analysis within all three computational schemes suggests that the interacting,
infrared-stable fixed point found previously collides with another fixed point
and becomes complex when the dimension is lowered from six towards five. Such a
result would conform to the expectation of triviality of $O(2)$ field theories
above four dimensions.
