Interestingly though GET with data exists in the wild, and has for many years.

I manage a http library class, and a customer encountered an API that required a GET but with data. (think query parameters passed as XML).

I implemented that for the customer, and then implemented the reverse in the server class. I'm not going to say its used a lot, but it makes semantic sense.

Incidentally it also becomes true for DELETE which is another request typically without a body.

This is the first I've heard of QUERY though, so look forward to reading up on that.