I wouldnt suggest this is the case for FUN but I've been involved with some alt coin chains from 2014 and they altered or the Dev team maybe switched projects and  it was on offer to convert one holding to another via proof of burn.   Seems a very adventurous idea to me but apparently it can work out.    Also there was blockchain I read recently that is relying on that idea for transmission, apparently it helps the value overall by reducing coin supply so is considered far more efficient then all the cost that goes into the proof of work energy usage and block rewards given there.   I always liked proof of stake and I hope that takes off and I guess that helps FUN but its not yet here.
   I'm quite certain efficiency is a normal process of capitalism and free markets, at some point it will matter even in crypto I reckon it comes to bear some weight in deciding value obviously we aren't determined in that way right now as POS is active for many years on some blockchains and not highly valued.   The only alternative idea you have at present is to sell the FUN, withdraw BTC to here and rebuy it once here but I would not guess Binance is making that path any easier either.