I believe they will at some point. Once banks become smart enough about it, they will all offer free Lightning transactions. It will be many months yet before that will happen.
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Banks are not what's responsible for the high network fees, its the BRC20 Ordinals nonsense.
No doubt. But banks will either become part of the bitcoin ecosystem or die. On-chain fees will become too high to make any sense to use for everyday transactions. If a bank advertised, "FEE-FREEE BITCOIN TRANSACTIONS", that would be enough to lure in customers.