I have now gotten another callback from United Healthcare and have been told that the sleep study request should have been marked as "approved" after the peer-to-peer review between my doctor and their doctor, but that had somehow not happened at the time. Thus, it seems, they generated a communication to my doctor indicating that it had been rejected and every time that we tried to communicate with them, they kept tripping over the rejection and then finding ways to not actually fix it to match the results of the peer-to-peer review.
But I am now assured that it has been marked approved and that, further, the deadline for getting the study done has been extended to year-end since it has taken them two months to actually get me an approval, leaving only one month to get the study by the original deadline. These are all good things.
So I called the sleep clinic and told them the good news. They were happy to hear this, as they had called United Healthcare twice to try to find out what the problem was and had not gotten called back.
Sadly, the United Healthcare website where the clinic goes to look up approvals appears to not be working at the moment. They'll call me back tomorrow...