Why is my Stateless Session Bean not sending message in a transaction when the Connection, Session and MessageProducer is manually cached?

Question

I'm working with a JMS broker that where it is slow/expensive to create MessageProducers. Therefore, I've decided to cache the MessageProducer in a stateless Session Bean. This works perfectly fine for non-transactional cases, but I'm having trouble when using Container Managed XA Transactions - the message is sent, but not as part of a transaction. Any idea as to what is wrong?

Problematic code:

@Stateless(name = "MyProducerSessionBean")
@TransactionManagement(value = TransactionManagementType.CONTAINER)
@TransactionAttribute(value = TransactionAttributeType.REQUIRED)
public class MyProducerSessionBean implements MyProducerSessionBeanLocal
{
    @Resource(name = "java:/jms/myConnectionFactory")
    ConnectionFactory myConnectionFactory;

    @Resource(name = "java:/jms/myOutputQueue")
    Queue myOutputQueue;

    Connection connection;
    Session session;
    MessageProducer producer;   

    public MyProducerSessionBean() {}

    @PostConstruct
    public void ejbCreate() {

        try {
            connection = myConnectionFactory.createConnection();
            session = connection.createSession(true, Session.AUTO_ACKNOWLEDGE);
            producer = session.createProducer(myOutputQueue);
        } catch (JMSException e) {
            cleanup();
            throw new EJBException(e);
        }
    }

    @TransactionAttribute(value = TransactionAttributeType.REQUIRED)
    @Override
    public void sendMessage() {
        try {
            ObjectMessage message = session.createObjectMessage("test");
            producer.send(message);
        } catch (JMSException e) {
            cleanup();
            throw new EJBException(e);
        }
    }

    @PreDestroy
    public void ejbRemove() {
        cleanup();
    }

    public void cleanup() {
        try {
            if(producer != null) producer.close();
        } catch (JMSException e) {
            e.printStackTrace();
        }
        try {
            if(session != null) session.close();
        } catch (JMSException e) {
            e.printStackTrace();
        }
        try {
            if(connection != null) connection.close();
        } catch (JMSException e) {
            e.printStackTrace();
        }
        producer = null;
        session = null;
        connection = null;
    }
}

Everything works fine when I create the connection inside the sendMessage() method, instead of caching the producer in @PostConstruct:

@Stateless(name = "MyProducerSessionBean")
@TransactionManagement(value = TransactionManagementType.CONTAINER)
@TransactionAttribute(value = TransactionAttributeType.REQUIRED)
public class MyProducerSessionBean implements MyProducerSessionBeanLocal
{
    @Resource(name = "java:/jms/myConnectionFactory")
    ConnectionFactory myConnectionFactory;

    @Resource(name = "java:/jms/myOutputQueue")
    Queue myOutputQueue;

    public MyProducerSessionBean() {}

    @TransactionAttribute(value = TransactionAttributeType.REQUIRED)
    @Override
    public void sendMessage() {
        try {
            Connection connection = myConnectionFactory.createConnection();
            Session session = connection.createSession(true, Session.AUTO_ACKNOWLEDGE);
            MessageProducer producer = session.createProducer(myOutputQueue);

            ObjectMessage message = session.createObjectMessage("test");
            producer.send(message);

            producer.close();
            session.close();
            connection.close();
        } catch (JMSException e) {
            e.printStackTrace();
            throw new EJBException(e);
        }
    }
}