  For asymptotically flat spacetimes, a conjecture by Strominger states that
asymptotic BMS-supertranslations and their associated charges at past null
infinity $\mathscr{I}^{-}$ can be related to those at future null infinity
$\mathscr{I}^{+}$ via an antipodal map at spatial infinity $i^{0}$. We analyse
the validity of this conjecture using Friedrich's formulation of spatial
infinity, which gives rise to a regular initial value problem for the conformal
field equations at spatial infinity. A central structure in this analysis is
the cylinder at spatial infinity representing a blow-up of the standard spatial
infinity point $i^{0}$ to a 2-sphere. The cylinder touches past and future null
infinities $\mathscr{I}^{\pm}$ at the critical sets. We show that for a generic
class of asymptotically Euclidean and regular initial data,
BMS-supertranslation charges are not well-defined at the critical sets unless
the initial data satisfies an extra regularity condition. We also show that
given initial data that satisfy the regularity condition, BMS-supertranslation
charges at the critical sets are fully determined by the initial data and that
the relation between the charges at past null infinity and those at future null
infinity directly follows from our regularity condition.
