Your custodial wallet provider sees all your transactions and can freeze your funds - so can your LSP.

I really like that (absent monitoring) that there is no permanent track record on chain, also we will produce on chain opening and closing tx that can tell a lot. I come to the conclusion that two things are necessary for functioning LN. 1) on chain privacy so you can open and close channels without revealing things 2) lower fees/more tx so force closes can not become an economic viable attack There are still plenty of drawbacks, but LN would already work much better on XMR than on BTC.