  This paper, in addition to our recent works, intends to explore the behavior
of the Wetterich flow equations in the portion of the theory space spanned by
non-branching melons constrained with Ward-identities. We focus on a rank-5
just-renormalizable tensorial group field theory and consider a non-trivial
extension of the local potential approximation namely effective vertex
expansion for just-renormalizable quartic melonic interactions, disregarding
effects coming from disconnected interactions. Investigating the dynamically
constrained flow, we show explicitly that results weakly rely on the number of
quartic interactions involved in the classical action. In particular, the
predictions for the fully connected model are essentially the same as for the
single colored model. Finally, closing the flow equations using Ward identities
without additional assumptions to compute integrals involved in the effective
vertex expansion, we do not find reliable fixed point in the unconstrained
theory space connected with the Gaussian region.
