You can certainly ask.
You say yourself the program is very desirable. Surely you are not the only one they admitted, and they have a whole list of backup applicants that have been waiting with bated breath for someone to reject the offer. Now that you have, the school would try to promptly notify the person on the waitlist, and that person would reply very quickly that, yes, they'd love to take up the offer. From what I recall, PhD applications are very competitive at the top schools and the stakes are high for applicants.
If the waitlister did indeed reply and and accept the offer, then your ship has definitely sailed. The university cannot go to the person they just said they'd admit, and say, "whoops, nevermind, we're not admitting you after all, go ask the other school you just rejected to take you back". The outrage and loss of reputation would be far greater than telling you "sorry, should have thought about that before rejecting our offer" (though they'll use much nicer language than me, I'm sure).
The offer of admission, and acceptance of that offer (or rejection for that matter) is often considered legally binding. While it rarely makes sense for anyone to actually go to court over something like this, in practice that adds additional gravity and friction to the decision. That is to say, there is a very narrow time window from the time you reject their offer to the time the next applicant on the list takes your spot, and if you miss that window you are almost certainly out of luck. The best you can do is hope to be waitlisted, but even that is doubtful - both due to time, the fact that universities often dislike applicants who reject them, and also that in rejecting their offer you reduce your own desirability as a candidate (who knows, if you changed your mind once you can change it again).
What if you hit this very narrow window? There's two possibilities:
- If you tell them you changed your mind after they send the email to the waitlister, but before the waitlister replies, I would say your chances are almost as slim. As I said, the offer cannot be easily withdrawn, unlike your asking to renegotiate, which can very easily be refused with no real consequence.
- If you tell them before they take any action, ie. before they tell the waitlister, then you might have a chance. It comes down to whether that particular program will hold your indecision against you. Some are understanding, because everyone knows the decision is hard and stressful. Some don't want students with even the slightest hint of not being absolutely committed.
On the one hand, you let them know on the 15th, so even the deadline has passed. In theory, all the decisions should have been finalized by now and it's too late for any do-overs. In practice, there's always a few programs that are late, I've heard of people on waitlists linger on into May and then get an offer after all. In addition, it was technically the weekend, so if you send an email now, they should see it first thing in the morning tomorrow... Assuming they don't decide to keep working over the weekends - again, something which while a huge factor, you cannot possibly know.
So in conclusion, it's a bit of gamble with long odds. Can you ask? Sure you can. Should you ask? Well you don't really lose much, so I'd say go ahead and try. Hell go buy a lottery ticket too while you're at it, you never know. But I'd say, whatever you do, don't go to the school you have accepted instead and tell them you won't be attending until you've got a concrete yes from the one you want.