o why not just withdraw to your wallet and then swap out using Boltz, SideShift, or FixedFloat, as you say?
Withdrawing to Muun is exactly withdrawing to one of my wallets, with the benefit that, unlike Boltz etc., swapping out of Muun doesn't incur a percentage fee.
You make a good point about the CEX knowing I bought Bitcoin and at what price, but erasing the KYC trace means they don't know what happened to it afterwards; in particular, they can't know if I still own it. If I withdraw from Kraken via LN, they won't even know I withdrew to the same wallet every time. For all the know maybe it wasn't even a withdrawal into my wallet, but a payment for goods or services, right after I bought the BTC, and the price I spent it at was roughly the same as the price I bought it at (and therefore no capital gains to pay tax on). While in reality my coins sit secretly in my cold storage and do a 10x over the next halving cycle.
If you want a no-KYC option cheaper than RoboSats etc., Relai.ch charges 1-2.5% and doesn't require KYC up to a certain amount.
Good points. I am not an expert on Lightning but the CEX (Kraken) knows you withdrew to Muun, right? (It can see the path and where the Lightning ended – someone correct me if I am wrong).
Hence the CEX knows you withdrew to Muun, and then Muun knows your cold storage address (since they have a record of your incoming LN, swap, and outgoing tx).
If the data from the CEX and Muun both leak or are provided to 3rd parties, you might still be connected to your coins. But I understand that is highly unlikely, if at all possible.
I like to do at least one more hop in LN in order to obfuscate the movements of funds a bit more (e.g., withdrawing LN to my own node and then sending it to Breez to use Boltz for swapping – good thing is Breez has Boltz integrated). But this might be too much.
If your solution with Muun works, I guess it is fine. You are not keeping your coins on Muun so you do not need to worry how they do things, you only care that they work for a few minutes for you to make your transactions.
If Muun is using submarine swaps to imitate LN, it is their problem and the economics of transaction fees will either allow them to do it (and there is not problem then) or it will force them to change this practice.
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