  This paper is focused on the functional renormalization group applied to the
$T_5^6$ tensor model on the Abelian group $U(1)$ with closure constraint. For
the first time, we derive the flow equations for the couplings and mass
parameters in a suitable truncation around the marginal interactions with
respect to the perturbative power counting. For the second time, we study the
behavior around the Gaussian fixed point, and show that the theory is
nonasymptotically free. Finally, we discuss the UV completion of the theory. We
show the existence of several nontrivial fixed points, study the behavior of
the renormalization group flow around them, and point out evidence in favor of
an asymptotically safe theory.
