Cannot Depth-Sort ParticleSystem in 2D

I have various ParticleSystems in a 2D game that need to be rendered between layers of Sprites. However the ParticleSystems render in front of everything. The shader on the ParticleSystem is the Sprite/Default shader.

I have made sure the particles are emitted only along 2D axes. I have tried changing the renderQueue property of the ParticleSystem material. I have tried other shaders - none will allow my particles to fade out, have a transparent background and respect rendering by depth all at the same time. I have tried modifying the Sorting Fudge property in both directions. I have tried changing the sorting layer of the game object. I have tried all manner of Z values for the game object. I have tried all the different Render Modes of the ParticleSystem renderer. I have tried combining properties of different Shaders in the hope I will make a custom shader that will allow the ParticleSystem to render at a given depth.

Nothing works.

How can I get a ParticleSystem to render at a given depth?

(Caveat: the texture has transparency and the ParticleSystem textures need to fade out.)