For mathematical convenience initial data sets in numerical relativity are often taken to be conformally flat. Employing the dual-foliation formalism, we investigate the physical consequences of this assumption. Working within a large class of asymptotically flat spacetimes we show that the ADM linear momentum is governed by the leading Lorentz part of a boost even in the presence of supertranslation-like terms. Following up, we find that in spacetimes that are asymptotically flat, and admit spatial slices with vanishing linear momentum that are sufficiently close to conformal flatness, any boosted slice can not be conformally flat. Consequently there are no conformally flat boosted slices of the Schwarzschild spacetime. This confirms the previously anticipated explanation for the presence of junk-radiation in Brandt-Bruegmann puncture data.