  It is commonly accepted that general relativity is the only solution to the
consistency problem that appears when trying to build a theory of interacting
gravitons (massless spin-2 particles). Padmanabhan's 2008 thought-provoking
analysis raised some concerns that are having resonance in the community. In
this work we present the self-coupling problem in detail and explicitly solve
the infinite-iterations scheme associated with it for the simplest theory of a
graviton field, which corresponds to an irreducible spin-2 representation of
the Poincar\'e group. We make explicit the non-uniqueness problem by finding an
entire family of solutions to the self-coupling problem. Then we show that the
only resulting theory which implements a deformation of the original gauge
symmetry happens to have essentially the structure of unimodular gravity. This
makes plausible the possibility of a natural solution to the first cosmological
constant problem in theories of emergent gravity. Later on we change for the
sake of completeness the starting free-field theory to Fierz-Pauli theory, an
equivalent theory but with a larger gauge symmetry. We indicate how to carry
out the infinite summation procedure in a similar way. Overall, we conclude
that as long as one requires the (deformed) preservation of internal gauge
invariance, one naturally recovers the structure of unimodular gravity or
general relativity but in a version that explicitly shows the underlying
Minkowski spacetime, in the spirit of Rosen's flat-background bimetric theory.
