Should ADS-B airborne and surface squitter transponder registers both be updated regards air/ground status?
I don't have the 1090ES MOPS at hand here, but it makes sense to me that both should be updated. 

At the moment that the registers are updated, it is not necessarily clear whether an airborne or a surface squitter will be transmitted next. In the period between update of the register and transmission of its contents, the vertical status of the aircraft can change.