For those that don't know what Fractional Reserve is .
Now back to your question,
Option 1: (Most Likely Scenario)
At the moment , 1 Onchain BTC = 1 LN Note , this is all controlled in the LN Hub software
At some point the LN code could easily be modified so 1 Onchain BTC = 2 LN Notes , but instead of these Notes being now worth ½ , LN Hubs still sell it as worth 1 BTC.
Basically Doubling the Supply of Fake Offchain BTC (LN Notes) overnight, and there is nothing stopping them from multiples of 10 or more just like a bank used to be required to only loan out 10 times their total deposit amount.
LN Notes are nothing more than a Offchain Representation of the Value of a BTC as such that are no real barriers preventing a fractional reserve system, it is only 1 software update away.
Quote
In the past, savers looking to keep their coins and valuables in safekeeping depositories deposited gold and silver at goldsmiths, receiving in exchange a note for their deposit (see Bank of Amsterdam). These notes gained acceptance as a medium of exchange for commercial transactions and thus became an early form of circulating paper money. As the notes were used directly in trade, the goldsmiths observed that people would not usually redeem all their notes at the same time, and they saw the opportunity to invest their coin reserves in interest-bearing loans and bills. This generated income for the goldsmiths but left them with more notes on issue than reserves with which to pay them. A process was started that altered the role of the goldsmiths from passive guardians of bullion, charging fees for safe storage, to interest-paying and interest-earning banks. Thus fractional-reserve banking was born.
Now back to your question,
Option 1: (Most Likely Scenario)
At the moment , 1 Onchain BTC = 1 LN Note , this is all controlled in the LN Hub software
At some point the LN code could easily be modified so 1 Onchain BTC = 2 LN Notes , but instead of these Notes being now worth ½ , LN Hubs still sell it as worth 1 BTC.
Basically Doubling the Supply of Fake Offchain BTC (LN Notes) overnight, and there is nothing stopping them from multiples of 10 or more just like a bank used to be required to only loan out 10 times their total deposit amount.
LN Notes are nothing more than a Offchain Representation of the Value of a BTC as such that are no real barriers preventing a fractional reserve system, it is only 1 software update away.
My understanding is that this is NOT how LN works. There are no "LN notes" as far as I understand. There are essentially only threats of broadcasting transactions that are exchanged. As far as I understand, an LN channel is nothing else but a growing stack of mutual threats to settle in the disadvantage of the cheater that are NOT broadcast, together with a "last correct balance" transaction, that can be broadcast by any of the parties when he wants to close the channel.
So there's not much of a chance to do fractional reserve banking that way, because there is no "IOU" (your "LN note") that is to represent a coin ; there is only a stack of mutually signed transactions one bricked into the other so that when broadcasting anything else but the last settlement, you automatically allow the counter party to take all your channel funds ; but without broadcasting them, they can't.
To the OP, as far as I understand, there is no "fractional reserve banking threat" by LN, but rather a "banking threat": namely that the dynamics of the LN is such, that it can only be beneficial to some central, big hubs to keep LN channels open, and that people have no choice but to lock in their funds to one of these central hubs. As such, these central hubs can apply AML/KYC rules, they can require the payment of fees of all kinds, they can censor certain of your transactions, and of course, they now all about every of your transactions, which they can report to authorities. But FRB is not part of it.
BTW, FRB is not applied any more in the current banking system, because there are no "reserves". The central bank reserves are nothing else but buffers for inter-bank settlements, which can be augmented at will by any bank.