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While with new bitcoin users the TX fee will increase, there is also a counterforce, the avoidance of the fee, by using offchain ledgers.
I also fear that the offchain transactions could make bitcoin pseudo-fiat, which could then create many fractional reserve scammer exchanges/wallets with their own offchain payment systems, and render bitcoin unimportant and creating a fiat currency of it
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I don't think that what you fear will materialize on a general level. To become an universal pseudo-fiat, the vast majority of (offchain) transactions would have to be processed by a single payment processor, that practices fractional reserve scam. Currently there is no indication that such a player is emerging. I think there will always be some attempts at implementing a fiat-scam, but these will most likely be of smaller magnitude and will not determine the general use case of Bitcoin.
In addition with side-chains / lightning, new ways are being developed to bundle small transactions together and ultimately settle them on the blockchain. This way, fees are saved and sufficient security against fractional reserve scams is ensured.
ya.ya.yo!