It happens all the time, unfortunately. People have been arrested trying to bring home pieces of sunken American warships from around Guadalcanal and New Guinea (because, as any warbird fan will say with despair, the US Navy holds claim to any of its property downed anytime, anywhere), and some of our sunken warships around Java, such as the USS Houston, have been picked over by covert scrappers. Same is true for our old facilities in the Philippines, like Corregidor.
I’m a sailor, I know, but the government doesn’t go looking for, nor bringing up, the remains.