Barium sulfate is a fine powder. By itself it isn't a coating. It is applied to paper in a suspension of either gelatin or casein, along with various other ingredients. There isn't a surface that is "only baryta."
I also think that it's extremely important to emphasise that industrially coated papers of the sort that @Federica seems to be talking about were subjected to supercalendaring and will all have a degree of gloss in their raw form - it's the matting of subsequent layers that affects the surface finish. Ilford's gloss and matte finish FB papers share the same base.
I've coated a lot of Baryta inkjet paper with no issues, it can take multiple soaks and looks beautiful. When you do an initial soak for wet coating, you can see there's a tiny bit of milkiness in the tray water, but nothing to suggest the coating has come off. I've been coating a layer of hardened gelatine followed by a layer of emulsion.