  We discuss nonparametric mixing distribution estimation under the
Gaussian-smoothed optimal transport (GOT) distance. It is shown that a recently
formulated conjecture -- that the Poisson nonparametric maximum likelihood
estimator can achieve root-$n$ rate of convergence under the GOT distance --
holds up to some logarithmic terms. We also establish the same conclusion for
other minimum-distance estimators, and discuss mixture models beyond the
Poisson.
