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.
Discussion
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.
Eventually the mempool will clear again once enough people use lightning wallets, or other Layer 2 solutions. I don't think the ordinals insanity will continue on forever.