No, the layer 2 inherits the trust-assumptions and security of the L1.
For instance, with LN you have to be online to secure your funds, otherwise you might lose them.
L2 only inherits the security of L1 when it settles, which by definition is not that often (that's the whole point!).
The "whole point" of L2 is to reduce the amount of resources used to settle its transactions on L1. Some L2s, like Lightning Network, achieve this by settling infrequently. But other L2s, like roll-ups, achieve it by batching transactions in clever ways so that less L1 resources are used on a per-transaction basis.
It could write to L1 immediately, but your writes can still not be trusted or right.
https://www.paradigm.xyz/2021/01/almost-everything-you-need-...