  We show that the BMS-supertranslations and their associated supermomenta on
past null infinity can be related to those on future null infinity, proving the
conjecture of Strominger for a class of spacetimes which are
asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical
3-manifold of both null and spatial directions of approach towards spatial
infinity, we impose appropriate regularity conditions on the Weyl tensor near
spatial infinity along null directions. The asymptotic Einstein equations on
this 3-manifold and the regularity conditions imply that the relevant Weyl
tensor components on past null infinity are antipodally matched to those on
future null infinity. The subalgebra of totally fluxless supertranslations near
spatial infinity provides a natural isomorphism between the
BMS-supertranslations on past and future null infinity. This proves that the
flux of the supermomenta is conserved from past to future null infinity in a
classical gravitational scattering process provided additional suitable
conditions are satisfied at the timelike infinities.
