Once we have eth1 inside of eth2 proper, then there’s no cost in allowing two-way convertibility of ETH ↔ BETH. At that point, we could just add a simple fee-payment system where fee-payers sign BLS signatures of [data_root, shard, slot, fee], and then the block proposal object includes the fee and the signer validator ID and the signatures of the fee-payers get aggregated into the signatures of the headers themselves.