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Buoyant forces are a bit tricky, and the recent thread here has clearly shown. I think I would look at this from the POV that gravity is still acting on the object, as mg, but the support force that comes from the pressure differences is distributed over the surface of the object. That won't be measured by a bathroom scale in contact with the object, but it will be reflected by a lager reading on the bathroom scale that is supporting the water tank in which the object is immersed. So, I guess that I would not call the object in buoyant equilibrium weightless (but I'm still thinking about it).
Weight as the reading on the bathroom scale may not be appealing when it is applied on an object partially submerged. The object becomes "weightless" because of buoyancy.
If you prefer, the weight can still be Mg. (In this case, g is the centripetal acceleration due to the rotation of the Earth.) There is no freefall acceleration! Perhaps, Bartlett could consider this as apparent weight. (If I am the referee for this paper, that will be my
recommendation!)