Fails to inject qualifier-annotated bean created by producer in embedded jetty

public class Main {
  public static void main(String[] args) throws Exception {
    ServletContextHandler handler = new ServletContextHandler();
    handler.addEventListener(new Listener());
    handler.addServlet(Consumer.class, "/*");

    Server server = new Server(8080);
    server.setHandler(handler);
    server.start();
    server.join();
  }
}

@ApplicationScoped
public class Consumer extends HttpServlet {
  @Inject
  @BindingAnnotation
  private String s;

  public String t() {
    return "test: " + s;
  }

  @Override
  protected void doGet(HttpServletRequest req, HttpServletResponse resp) throws ServletException, IOException {
    resp.setStatus(HttpServletResponse.SC_OK);
    resp.setContentType("text/plain");
    resp.getWriter().write(t());
  }
}

@ApplicationScoped
public class Producer {
  @Produces
  @BindingAnnotation
  public String get() {
    return "test";
  }
}