Recently, secure data deletion or data sanitization has been identified as a key technology of storage devices to securely delete obsolete sensitive data that are no longer used. However, secure data deletion requires extra management efforts on flash memory storage devices, due to the deferred reclamation of flash blocks in many flash translation layer schemes. The emerging zoned namespace storage further exacerbates the design complexity of secure data deletion, due to the much larger size of a zone than that of a flash block. Concerning the very long latency to reset an entire zone, once some data have been written into a zone, it is very difficult to securely delete them from the zone. To achieve efficient and secure data deletion on zoned namespace storage, we propose persistence relaxation for efficient and secure sanitization (PRESS), which considers the working principle of zones of zoned namespace storage and allows the fine-grained control of deferred data persistence. As a result, applications can efficiently delete their recently written data or make the data persistent for long-term storage. Our proposal, PRESS, is evaluated through a series of experimental studies, where the results are quite encouraging.