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> If they get caught cheating, they lose their stake

in that chain. In any distributed blockchain there's going to be several chain(tips) at any given time.

Issuing staking transactions on more than one chain is one of the defined slashing conditions. All the transactions are public so any other staker can see that you've done it, submit the evidence in a transaction of their own, and earn a reward while causing some or all of your stake to be destroyed.

("Some or all" because the devs recognize that you might get hacked, so the penalty depends on how much of the total stake misbehaves at once. If it's only a small portion, it's not damaging and presumed to be accidental.)

What if the misbehaving actor refuses to extend the chain that includes evidence of its own cheating?

Ie. the cheater just extends the chain at one block before the block that contains evidence of the cheating.

How will clients agree to use the chain that contains the cheating proof, and not the other — just as valid — chain?

You are talking about the nothing at stake problem. It's address here: https://github.com/ethereum/wiki/wiki/Proof-of-Stake-FAQs#wh...

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