  The thermodynamic relation between pressure and density (i.e. the equation of
state) of cold supranuclear matter is critical in describing neutron stars, yet
it remains one of the largest uncertainties in nuclear physics. The extraction
of tidal deformabilities from the gravitational waves emitted in the
coalescence of neutron star binaries, such as GW170817, is a promising tool to
probe this thermodynamic relation. Equation-of-state insensitive relations
between symmetric and antisymmetric combinations of individual tidal
deformabilities, the so-called "binary Love relations", have proven important
to infer the radius of neutron stars, and thus constrain the equation of state,
from such gravitational waves. A similar set of relations between the moment of
inertia, the tidal deformability, the quadrupole moment, and the compactness of
neutron stars, the so-called `I-Love-Q' and `C-Love' relations, allow for
future tests of General Relativity in the extreme gravity regime. But even the
most insensitive of such relations still presents some degree of
equation-of-state variability that could introduce systematic uncertainties in
parameter extraction and in model selection. We here reduce this variability by
more than 50% by imposing a prior on the allowed set of equations of state,
derived from the posteriors generated from the analysis of GW170817. The
resulting increase in insensitivity reduces systematic uncertainties in the
extraction of the tidal deformability from future gravitational wave
observations, although statistical uncertainties currently dominate the error
budget, and will continue to do so until the era of Voyager-class detectors.
