  The process whereby the supermassive black holes populating the centers of
galaxies have been assembled remains to be established, with the relative
importance of seeds provided by collapsed Population-III stars, black holes
formed in nuclear star clusters via repeated mergers, or direct collapses of
protogalactic disks yet to be determined. In this paper we study the prospects
for casting light on this issue by future measurements of gravitational waves
emitted during the inspirals and mergers of pairs of intermediate-mass black
holes, discussing in particular the roles of prospective measurements by LISA
and the proposed atom interferometers AION and AEDGE. We find that, the
expected number of detectable IMBH binaries is $O(100)$ for LISA and AEDGE and
$O(10)$ for AION in low-mass seeds scenarios and goes down to $O(10)$ for LISA
and below one for AEDGE and AION in high-mass seed scenarios. This allows all
of these observatories to probe the parameters of the seed model, in particular
if at least a fraction of the SMBHs arise from a low-mass seed population. We
also show that the measurement accuracy of the binary parameters is, in
general, best for AEDGE that sees very precisely the merger of the binary.
