How do I get a SessionScoped CDI bean from inside a Filter?

This question is related to a previous one on writing a session timeout handler.

The answer in that thread involved accessing various session-scoped managed beans from the servlet. The recommendation (as seen here) is to do this in the filter:

HttpSession session = request.getSession(false);
User user = (session != null) ? (User) session.getAttribute("user") : null;

Presumably this fetches a session bean of class User. The problem is this doesn't work.

What goes wrong are that the beans are there in the session attributes, but they are wrapped by Weld facilities. I wrote the doFilter() method as follows:

public void doFilter(ServletRequest request, ServletResponse response,
        FilterChain chain)
        throws IOException, ServletException {
    HttpServletRequest req = (HttpServletRequest) request;
    HttpServletResponse res = (HttpServletResponse) response;

    String sp = req.getServletPath();
    System.out.println("------------------------");
    System.out.println("doFilter(): " + sp);

    if (!sp.startsWith("/javax")) {  // eliminates many requests
        HttpSession session = req.getSession();
        Enumeration<String> en = session.getAttributeNames();
        int count = 0;            
        while (en.hasMoreElements()) {
            String e = en.nextElement();
            System.out.println("Attribute " + ++count + ": " + e);
        }
    }
    chain.doFilter(request, response);
}

When this dumps out the session attributes, I typically get something like this:

INFO: ------------------------
INFO: doFilter(): /Display.xhtml
INFO: Attribute 1: org.jboss.weld.context.http.Htt