How does L2 works ? Like matic , ARB or OP

I want to keep some usdt in ledger or trezor bit I don’t wanna keep them with eth bc there is a transaction charge of 10-15$ for every transaction in and out.

So my question is what happens to my usdt stored on L2 if that chain is closed permanently or comes out as scam such as LUNA ( stable on it went to 0).