Try and make a test with the acrylic gesso on glass or metal, when it is completely dry, it takes long time to get rest water out, then brush, or with the side of a finger, a little water over the surface to see if it sinks in, in most cases I think it will, so linseed and sunflower oil also will. I would also test the oilground itself on glass.
I sometimes apply a very thin coat of a pure acrylic medium over the gesso, and if not, then medium out with a mix of linseed oil and a pure alkyd. The polymerised oil will not sink in or to a much lesser extent, so therefore it is good to have a little in a medium.