Sadly it doesn't exhibit the banding that real legit lorentz-invariant propagators do. It has some banding, true, but it wibbles in the wrong direction.
Oh wait no! If I adjust the parameters a bit, I do see some nice properly oriented banding.
Maybe I was just losing it in the Moiré before. Dunno if it's actually properly hyperbolic or not.
And as just one further thought, surely this picture is not Lorentz *invariant* because you can see a band of roughly constant value down the center vertically, which is a dead giveaway of a privileged rest frame. But maybe you can make it Lorentz *covariant* if you apply some transformation on the relative amplitudes of the 'L' and 'R' particles at the same time?