Deviations from the Poisson distribution are exactly what he is testing for. The paper you referenced discusses how to model in the presence of overdispersion. Not how to test that as a hypothesis.
Moreover Poisson distributions are generated from events that occur in a very small fraction of all opportunities, but with many opportunities each time period. So, unless the timing of shootings is correlated (again, the thing he is testing for), this is a perfect use case for the Poisson distribution.
Moreover Poisson distributions are generated from events that occur in a very small fraction of all opportunities, but with many opportunities each time period. So, unless the timing of shootings is correlated (again, the thing he is testing for), this is a perfect use case for the Poisson distribution.