I have sprayed OB eggshell with my HVLP. I thinned it with white spirit (about 10%) and added some Owatrol oil. Be advised that the over spray is very sticky and remains so for quite a long time. I wouldn't consider spraying it indoors (unless it was a building site).
I have never tried to spray WB eggshell. I suspect that my old Fuji HVLP would struggle to atomise it unless I thinned it so much that the paint end up being compromised (given that waterbased paints cure via coalescence).
Most of my furniture/cabinet spraying is with acid cat (2K) paint. It is a 2 pack paint that is very easy to atomise. It is also far more durable than oil or waterbased paints. I can spray a sheet of MDF and lift it up after as little as 30 minutes. Depending on the amount of pigment in the paint, I can sand it after a couple of hours. On the down side, it is isocyanate based and very toxic. You are supposed to have a clean air breathing supply when working with it. I admit that I normally use it outdoors whilst wearing an ABEK face mask, far from ideal and not something I would ever recommend.
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