Hi Shunsai,
Block approvers own >50% of the stake. If they approve blocks containing conflicting approvals, that will invite many forking related attacks on the chain including double spend. Approvers can approve another block containing all valid approvals, so they don't benefit by violating the rule. Would a rational stakeholder approve a block that may lead the devaluation of their asset without providing them any benefit? I think not. (This has the same basis as PoS.)
What I mean was the block creators benifit from violating the rule, not the approvers. They could receive more approvals from using both unconflictful and conflictful contents. Isn't that right? Regardless of the benifit anyway, why does anyone doing their duty when there is no evidence and no punishment for not doing so? What if somebody use an incomplete mining application?
In absence of >50% approvals, no new blocks will be created in those slots. It may not sound that good at first but this is the best policy. It's preferable to not process any transactions than to risk the entire blockchain. Once the connectivity is restored, the stakeholders will approve (or reapprove) the set of blocks offered on top of the previously approved block.
As a matter of experience, >50% online stake is very unusual. The problem of getting stuck is critical because it could happen anytime and last a very long period of time, not just under network separation. Shouldn't you find a way to deal with that?
All block based chains like Proof-of-Approval including Bitcoin and Ethereum, have the so called probabilistic finality. The newest 'n' blocks in the chains are not considered stable. For Bitcoin, users want that 'n' to be 6 blocks or more. For Proof-of-Approval, it is just 1 block.
In this respect, the word "stable" is meaningless when the onchain data could still be changed and that's why Bitcoin and Ethereum are called "nonfinality". Protocols like PBFT and Casper provide finalites by the collaboration of Valiators but I didn't see "finality" in your protocol.