I’d passed this up the chain to folks with more control over the site’s backend. It’s pretty much confirmed that it’s a cookie discrepancy with https/http. This can happen if you have older bookmarks that point to the http URL instead of the https ones. Your session cookie could get bound to the http URL and when the site redirects to https, it causes confusion and thinks you’re logged out.
That’s what’s most likely the issue we’re seeing here. An effective solution would be to just disable http (including redirects to https) and only use https, but that will cause a lot of people’s bookmarks and links to BA break… so I’m not sure that’s really the best solution overall.
We’re still discussing this, but for now, the best solution for members of the forum who still have this problem even after updating bookmarks is to use one of the extensions mentioned in Tim Tylor’s post.