Calling the Lightning Network (LN) a Proof-of-Stake (PoS) system is orthogonal to the concept usually labeled as PoS.
What we usually describe as PoS is a mechanism to achieve distributed consensus among a network of participants. Staking is hereby providing updates to the whole network's state, by which participants earn fees for contributing to the validity and the security of the network.
On the other hand, there is no global consensus in the LN. The LN is a system of bilateral smart-contracts, where you may earn a fee by providing liquidity to other users. Nodes along a payment path forward payments, but even these contracts are created, verified, and executed between a small number of participants. Conflict resolution is achieved by publishing on-chain transactions which indirectly leverages the proof-of-work of the Bitcoin network. While nodes announce themselves and their channels to their peers, there is no datum on which the whole LN must agree.
In fact, not even the two channel owners locally run a consensus protocol. If one side proposes an HTLC, the other side can accept or decline it. When the channel owners disagree on something, e.g. feerates, or collaboration breaks down in another manner, the process breaks down. They will not reconcile, but shut down the channel and disconnect rather than converging on a shared state.
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