  If the progenitor of GW170817 harbored a pulsar, then a Poynting flux
dominated bow-shock cavity would have been expected to form around the
traveling binary. The characteristic size of this evacuated region depends
strongly on the spin-down evolution of the pulsar companion, which in turn
depends on the merging timescale of the system. If this evacuated region is
able to grow to a sufficiently large scale, then the deceleration of the jet,
and thus the onset of the afterglow, would be noticeably delayed. The first
detection of afterglow emission, which was uncovered 9.2 days after the
$\gamma$-ray burst trigger, can thus be used to constrain the size of a
pre-existing pulsar-wind cavity. We use this information, together with a model
of the jet to place limits on the presence of a pulsar in GW170817 and discuss
the derived constraints in the context of the observed double neutron star
binary population. We find that the majority of Galactic systems that are close
enough to merge within a Hubble time would have carved a discernibly large
pulsar-wind cavity, inconsistent with the onset timescale of the X-ray
afterglow of GW170817. Conversely, the recently detected system J1913+1102,
which host a low-luminosity pulsar, provides a congruous Milky Way analog of
GW170817's progenitor model. This study highlights the potential of the
proposed observational test for gaining insight into the origin of double
neutron star binaries, in particular if the properties of Galactic systems are
representative of the overall merging population.
