Some pages may be stored on the device if the project has caching configured. This does not guarantee that all content or actions work offline. Payments, current data and external services usually need a connection; the specific behaviour should be stated in the proposal.
After an update is published, the version shown may depend on the browser and its saved files. If you see old content, reopen the site online and refresh the page. Notifications, background synchronisation and full offline operation are separate features and should not be assumed simply because a site is a PWA.
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