I would say May 20th is a little late, only because applicants have to give final decisions by May 15th. Schools will be expecting huge shuffles on that day, so I'm assuming they'll have prepared at least a preliminary order or list of who they might want to take off the waitlist. If I was on a waitlist, I would probably send my LOI some time in April, maybe mid to late month. But that's just me, I haven't really read about LOI successes a lot from previous years and other waitlists.
Honestly I feel the same way as others do on this thread, really hoping for an eventual acceptance for you here! I wish there were a way to give specific waitlisters your spot!! If I decide on another school at some point in this cycle I would definitely want my spot to go to you 🙂
best of luck, we're all rooting for you!