  Neutron star mergers (NSMs) are promising astrophysical sites for the rapid
neutron-capture ("$r$-") process, but can their integrated yields explain the
majority of heavy-element material in the Galaxy? One method to address this
question has utilized a forward approach that propagates NSM rates and yields
along with stellar formation rates, in the end comparing those results with
observed chemical abundances of $r$-process-rich, metal-poor stars. In this
work, we take the inverse approach by utilizing $r$-process-element abundance
ratios of metal-poor stars as input to reconstruct the properties---especially
the masses---of the neutron star (NS) binary progenitors of the $r$-process
stars. This novel analysis provides an independent avenue for studying the
population of the original neutron star binary systems that merged and produced
the $r$-process material incorporated in Galactic metal-poor halo stars. We use
ratios of elements typically associated with the limited-$r$ process and the
actinide region to those in the lanthanide region (i.e., Zr/Dy and Th/Dy) to
probe the NS masses of the progenitor merger. We find that NSMs can account for
all $r$-process material in metal-poor stars that display $r$-process
signatures, while simultaneously reproducing the present-day distribution of
double-NS (DNS) systems. However, the most $r$-process enhanced stars (the
$r$-II stars) on their own would require progenitor NSMs of very asymmetric
systems that are distinctly different from present ones in the Galaxy. As this
analysis is model-dependent, we also explore variations in line with future
expectation regarding potential theoretical and observational updates, and
comment on how these variations impact our results.
