Since different browsers means different computers, it's quite likely the user doesn't want their hotkeys to sync across browsers (e.g. wants different hotkeys for work and home computers). It's particularly important for if the user is running on different OSes -- then we have to make sure the hold key setting isn't syncing.
Maybe the best solution would be to have an option for whether to sync across browsers, with default to no.
This should be pretty simple to do; it's just a matter of using local storage instead of chrome.sync. Might be a bit tricky with the option to do either.