Anyone who has been following these posts for a while has noticed the running thread of limitations caused by Android’s WebView, which is why the 4.x series is going to include a forked version called Privacy WebView. While developing tabbed browsing I discovered that first-party cookies are enabled/disabled by app, while third-party cookies are enabled/disabled by WebView. This means that if the current tab enables cookies, they will also be enabled for any background requests made by any other tabs that are currently open.
There is nothing I can do about this presently, but it will be addressed in the 4.x series. In the meantime, I have added a warning in the settings.