Does nonlocal gravity yield divergent gravitational energy-momentum fluxes?

Energy-momentum conservation requires the associated gravitational fluxes on an asymptotically flat spacetime to scale as $1/r^2$, as $r \to \infty$, where $r$ is the distance between the observer and the source of the gravitational waves. We expand the equations-of-motion for the Deser-Woodard nonlocal gravity model up to quadratic order in metric perturbations, to compute its gravitational energy-momentum flux due to an isolated system. The contributions from the nonlocal sector contains $1/r$ terms proportional to the acceleration of the Newtonian energy of the system, indicating such nonlocal gravity models may not yield well-defined energy fluxes at infinity. In the case of the Deser-Woodard model, this divergent flux can be avoided by requiring the first and second derivatives of the nonlocal distortion function $f[X]$ at $X=0$ to be zero, i.e., $f'[0] = 0 = f''[0]$. It would be interesting to investigate whether other classes of nonlocal models not involving such an arbitrary function can avoid divergent fluxes.