When you mention that the aircraft cabin pressure climbs with greater altitudes, my first guess would be that you've got a leak. Most likely a big one.
The packs can only output so much. Beyond a certain altitude..... they just can't keep up.
If ground time allows, I'd put somebody inside and take it to 3 or 4 PSI. A couple of guys can check the outside for holes, blown ducts, or bad door seals. If you have a leak, I bet you can probably hear it.
If the outflow valve is manually closed on the ground, can you verify that it is fully closed visually?
Did the crew noticed if the outflow valve was fully closed in flight?
I had a
a while back. The packs could keep up, but the system warned of high pack inflow.
If time doesn't allow..... I'd ask the crew if they could try to bring the cabin down manually on the next flight..... If they can, the problem lies elsewhere.