By the way: the URL you posted responds with “invalid screencast ID”.
A user specific time zone is not possible because of the way how Timed Content works:
- The browser requests a page using HTTP.
- On the server Timed Content will handle all shortcodes in the page and will either hide or show content for server side rules or server side shortcodes according to the rule definition.
- WordPress will send the final page as response to the browser.
The problem: HTTP requests do not contain any client timezone information, so WordPress does not know what timezone the user is in. That’s the reason why the timezone must be part of the rule and is the same for all users who visit the website.