the cabin will climb at the same rate as the aircraft
.. doesn't follow as the altimeter deltas are linear while the pressure deltas are not.
Unless you want to do the sums to get the actual figures, it might be better to say things like -
(a) with a given pressurisation setting commanded, the system will follow the system schedule for cabin altitude with increasing/decreasing aircraft altitude so long as the diff is less than max.
(b) at max diff an increasing/decreasing aircraft altitude will result in an increasing/decreasing cabin altitude