Using Post Process to sort specific materials

I’m looking to create a post processing effect that would be used to make one material render in front of another specific material without otherwise affecting their render order with everything else in the world.

What I want to do is make it so the white panel shows up in front of the grey panel but both of them are still behind the blue panel.

I’ve been told it’s possible to make a material render on top of another material using post process effects. Is it possible to do that without making it render on top of everything else? Just on top of one designated material?

I’d appreciate some pointers on where to start with this, please.