Is it possible to inject a @Produces into an @EJB?

Let's say I have a SLSB like:

@Local
public interface MyService {
   void fetchWidget(Long id);
}


@Stateless
public class MyServiceBean implements MyService {
   
   @Inject @MyData
   EntityManager em;

   public void fetchWidget(Long id) {
      return em.find(Widget.class, id);
   }

}

And a factory that declares my Producer:

@ApplicationScoped
public class EMFactory {

   @Produces
   @MyData
   @PersistenceContext(unitName="theUnit")
   static EntityManager entityManager;

}

Should this work assuming the @MyData annotation has been properly defined? I've tried, but haven't had any luck. Not entirely clear to me whether this is legal or not.

Rick