Make sure that BEEFY is working correctly on a testnet, with additional onliness assumptions (at least 2/3 + 1 validators online throughout the entire session length; i.e. no catch-up necessary).

The implementation should be based on the description in https://github.com/paritytech/grandpa-bridge-gadget/blob/td-docs/docs/beefy.md#lean-beefy, note that some of the issues might be a little bit outdated in description, but the high level description should provide all the details.