Could our moon have a submoon?

[Peter_Morris]

previous discussion:

Could a moon have a moon of its own?

I think there are no examples in our own Solar System. But suppose we imagine a Jupiter-like gas giant orbiting a faraway star. This has several moons, including an Earth-like ball. Could there be a Luna-like rock orbiting? Is there any reason why it couldn’t happen? Maybe the gravity of the main planet would prevent a stable orbit forming, or something.