What would a potential absence of power-law singularities in of N=4 SYM four particle amplitudes mean from a physics point of view?

Question

In this post, Lumo explains that Nima and friends have by applying modern twistor, Grassmannian, amplituhedron methods found out that four particle amplitudes of N=4 SYM are potentially free of powerlow singularities up to all orders in perturbation theory.

In the comments, Lumo says that this is potentially related to the UV finiteness of the theory, but not much can be said about the physics implications of this result at present.

But maybe some people here have some additional ideas about what these results could mean from a physics point of view, if true ?