Unimodular gravity with external sources
The only allowed source of the gravitational field in the unimodular theory, invariant under area-preserving (transverse) diffeomorphisms as well as Weyl transformations, is just the traceless piece of the energy-momentum tensor. This fact notwithstanding, the free energy produced by arbitrary sources (not only static ones) is identical to the one predicted by general relativity. This encompasses all weak field tests of gravitation.