Liquid is a permissioned sidechain where you can move Bitcoins into and out of it via a 2-way federated peg. To do this you would send Bitcoins to a special output on the Bitcoin blockchain, and then 'Liquid Bitcoins' would be created and issued to you on the Liquid sidechain - which you can then transact with faster confirmations and more privacy than on the Bitcoin mainchain. You must trust the block-signing federation to not collude to double-spend against you on the sidechain by creating more than one version on the chain (this is prevented in Bitcoin with PoW mining).

The mainstay concept is nothing to do with 2-way pegging, but is designed to remove the trust required in the block-signing federation - it transfers the trustless immutability of the Bitcoin blockchain to the sidechain. It secures the sidechain in a similar way to merge-mining, but without requiring the cooperation or permission of Bitcoin miners.

There is nothing stopping Liquid, or any other sidechain, from using this method to secure the chain.