This question sprouted from Arnold's comments of this post. I've seen more theoretical evidences hinting at the triviality of pure QED and pure $\phi^4$ in 4D spacetime than the opposite, I'll be more explicit:

(1)Perturbative expansions to a few loops order, for both QED and $\phi^4$. But I confess I myself have only verified the 1-loop results.

(2)Claims from lattice simulations. I never verified this for myself, and lattice $\phi^4$ numerics seem to be well cited, while the lattice QED numerics less so.

(3)Nonperturbative constructive results on $\phi^4$, a rigorous proof of triviality is said to be only a few lemmas away from completion (I remember this claim from an online lecture of Arthur Jaffe).

Now by "pure" I mean constructions that only make use of the infrared terms, for example if I remember correctly, evidence (3) can't exclude---even with the completion of the unproved lemmas---the possibility for the existence of a scalar field theory with additional $\phi^6, \phi^8, \phi \Box^2 \phi$ terms etc, such that it flows to a pure $\phi^4$ in the infrared.

What are the supporting evidences for the opposite claims?