If FOCIL includers don’t store (or query, or verify a proof) the validity-only state (balance and nonce basically), they can’t know which transactions should be kept or pruned in the mempool, which means an attacker can easily spam the mempool with a lot of invalid transactions, that would then end up in ILs and crowd out valid ones.
In practice they can definitely choose not to hold state if they don’t want to, but then they wouldn’t be able to participate in FOCIL and propose “good”, relevant ILs. If we don’t assume all nodes will want to participate in FOCIL (which is fair, it’s just a departure from the model we have today where all participating nodes running an EL client maintain the public mempool according to the exact same rules), then FOCIL should probably opt-in, and FOCIL nodes should be cleanly separated from other nodes performing other duties (e.g., Attester-Includer Separation) in the future. But then they might need incentives with either fees, or issuance. This is a topic we’re actively exploring but it’s very tricky find a simple, incentive compatible solution there.