  The present-day orbit distribution of the Neptune Trojans is a powerful probe
of the dynamical environment of the outer solar system during the late stages
of planet migration. In this work, I conservatively debias the inclination,
eccentricity, and libration amplitude distributions of the Neptune Trojans by
reducing a priori unknown discovery and follow-up survey properties to nuisance
parameters and using a likelihood-free Bayesian rejection sampler for parameter
estimation. Using this survey-agnostic approach, I confirm that the Neptune
Trojans are a dynamically excited population: at $>$95% confidence, the Neptune
Trojans' inclination width must be $\sigma_i > 11^\circ$. For comparison and
motivation purposes, I also model the Jupiter Trojan orbit distributions in the
same basis and produce new estimates of their parameters (Jupiter Trojan
$\sigma_i=14.4^\circ \pm 0.5^\circ$, $\sigma_{L11} = 11.8^\circ \pm 0.5^\circ$,
and $\sigma_e = 0.061\pm 0.002$). The debiased inclination, libration
amplitude, and eccentricity distributions of the Neptune Trojans are nominally
very similar to those of the Jupiter Trojans. I use these new constraints to
inform a suite of simulations of Neptune Trojan capture by an eccentric,
rapidly-migrating Neptune from an initially dynamically-hot disk. These
simulations demonstrate that if migration and eccentricity-damping timescales
were short ($\tau_a \lesssim 10$ Myr, $\tau_e \lesssim 1$ Myr), the disk that
Neptune migrated into must have been pre-heated (prior to Neptune's appearance)
to a width comparable to the Neptune Trojans' extant width to produce a
captured population with an inclination distribution width consistent with that
of the observed population.
