A320 Captain reported pressurization problems and being unable to rectify the issue with manual mode as per ECAM. After landing crew was unable to depressurize the aircraft on the ground until a cockpit window was opened after air conditioning packs were selected to off.

Narrative

Aircraft X Inbound to ZZZ at 24;000 ft. straight and level we started having a pressurization issue. We recognized the pressurization issue due to our ears popping and feeling pressure. We looked at the pressurization screen and seen the cabin altitude at negative 4;400 ft. and the VSI moving erratically up a down reaching negative 3;900 ft. at one point. This was followed by a master caution CAM CAB PR SYS 2 FAULT. The outflow valve showed completely closed; and the safety valve showed completely open. We followed the ECAM to manually operate the outflow valve by using the manual vertical speed. This however; did not work. We did an urgent decent from 24;000 feet to 8;000 ft. and [requested priority handling] straight to ZZZ. We continued to troubleshoot the problem with no success. Upon landing on Runway XXR in ZZZ; making a turn off; our ears were still popping and feeling pressure. The cabin altitude still showed negative 4;400 ft. and a differential pressure psi of 2.5. After turning both packs off this still did not fix the problem. I asked the First Officer (FO) if he could open his cockpit window to depressurize the aircraft. The First Officer was unable to open his window. I was finally able to open my window and depressurize the airplane. We continued to taxi to the gate. No injuries reported just inner ear discomfort from Crew and Passengers. We filled out the logbook and gave a full brief of our issue to maintenance and left the aircraft.