  The only binary neutron star merger gravitational wave event with detected
electromagnetic counterparts recorded to date is GRB170817A. This merger
occurred in a rarefied medium with a density smaller than $10^{-3}-10^{-2}~{\rm
cm}^{-3}$. Since kicks are imparted to neutron star binaries upon formation,
and due to their long delay times before merger, such low-density circum-merger
media are generally expected. However, there is some indirect evidence for
fast-merging or low-kick binaries, which would coalesce in denser environments.
Nonetheless, present astronomical data are largely inconclusive on the
possibility of these high-density mergers. We describe a method to directly
probe this hypothetical population of high-density mergers through
multi-messenger observations of binary neutron star merger afterglows,
exploiting the high sensitivity of these signals to the density of the merger
environment. This method is based on a sample of merger afterglows that has yet
to be collected. Its constraining power is large, even with a small sample of
events. We discuss the method's limitations and applicability. In the upcoming
era of third-generation gravitational wave detectors, this method's potential
will be fully realized as it will allow us to probe mergers that occurred soon
after the peak of cosmic star formation, provided the follow-up campaigns are
able to locate the sources.
