  It has been suggested that amplitudes for quantum higher-spin massive
particles exchanging gravitons lead, via a classical limit, to results for
scattering of spinning black holes in general relativity, when the massive
particles are in a certain way minimally coupled to gravity. Such limits of
such amplitudes suggest, at least at lower orders in spin, up to second order
in the gravitational constant $G$, that the classical aligned-spin scattering
function for an arbitrary-mass-ratio two-spinning-black-hole system can be
obtained by a simple kinematical mapping from that for a spinning test black
hole scattering off a stationary background Kerr black hole. Here we test these
suggestions, at orders beyond the reach of the post-Newtonian and
post-Minkowskian results used in their initial partial verifications, by
confronting them with results from general-relativistic "self-force"
calculations of the linear perturbations of a Kerr spacetime sourced by a small
orbiting body, here considering only results for circular orbits in the
equatorial plane. We translate between scattering and circular-orbit results by
assuming the existence of a local-in-time canonical Hamiltonian governing the
conservative dynamics of generic (bound and unbound) aligned-spin orbits, while
employing the associated first law of spinning binary mechanics. We confirm,
through linear order in the mass ratio, some previous conjectures which would
begin to fill in the spin-dependent parts of the conservative dynamics for
arbitrary-mass-ratio aligned-spin binary black holes at the fourth-and-a-half
and fifth post-Newtonian orders.
