  In this note, we revisit the recent work of Diakonikolas, Gouleakis, Kane,
Peebles, and Price (2021), and provide an alternative proof of their main
result. Our argument does not rely on any specific property of Poisson random
variables (such as stability and divisibility) nor on any "clever trick," but
instead on an identity relating the expectation of the absolute value of any
random variable to the integral of its characteristic function:
  \[
  \mathbb{E}[|X|] = \frac{2}{\pi}\int_0^\infty \frac{1-\Re(\mathbb{E}[e^{i
tX}])}{t^2}\, dt
  \]
  Our argument, while not devoid of technical aspects, is arguably conceptually
simpler and more general; and we hope this technique can find additional
applications in distribution testing.
