Yes, the suspended seating frame should be raised (using the carriage bolts) so the top of the frame is flush with the top of the cabinet or underside of the counter-top. This will ensure that the chairs will be the proper distance from the footrest and counter-top for convenience and comfort.
If the counter-top is designed with a bull-nose feature, the suspended seating frame should be lowered enough to clear the drop of the bull-nose. Please use care to not drop the frame too far, thus compromising the positioning of the chair height.