Not sure about sessionStorage, but I'm pretty sure that most browsers will prompt the user for confirmation (ie, a dialog saying "this site wants to...") when the code uses localStorage. Test carefully to make sure you don't spook your visitors.
Edit: Also this text seems to suggest you can't share data between windows, because they will be considered separate sessions. "Opening a page in a new tab or window will cause a new session to be initiated, which differs from how session cookies work."
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