A321 Captain reported a pressurization malfunction during a cruise climb. The Captain requested a lower altitude and continued to destination.

Narrative

DISPATCHED SINGLE-PACK (RIGHT PACK INOP). Level @ FL340; both Cabin Pressurization Page and Cruise Page appeared Normal. Approximately 35 nm South of ZZZ1; NNE bound; received clearance from ZZZ Center to climb to FL350. Upon reaching 350; observed CAB V/S indicating 'climb'; however; several minutes later CAB V/S still indicated 'climb' and CAB ALT appeared elevated (~>8000). Suspected a malfunction; F/O (Pilot Flying) concurred; advised ZZZ Center we might have a pressurization problem; requested lower altitude; initially given FL240. Shortly thereafter during descent; received EXCESS CAB ALT on/per ECAM. Exercised applicable procedures; [requested priority handling] with ZZZ Center; received clearance to 10;000 ft. Cabin altitude reached maximum of ~11;500 ft; passenger mask not deployed; cockpit crew 02 only. Subsequent to initial decent (~4 minutes); received ECAM CAB PR 1 + 2 FAULT; observed LEFT PACK FAILED/OFF on overhead. Communication with Flight Attendants advised all was well with both passengers and cabin. Weather both enroute and destination was good; plenty of fuel; thus ZZZ2 was suitable and prudent.Single-Pack Operation. Integrity Test of operable Pack Control Valve?