The problem is that in some situations the required deposit may be infinite.
Consider an example where Eve deposits $10, spends it in increments of 1 cent by paying to herself, and then exits all intermediate 1000 UTXOs potentially making $5,000
In this example, isn’t it true that 1000 separate fraud proofs would need to be submitted? If $1 gas fee is paid for each fraud proof, the total is $1000 of fees. So you would need to require a deposit of $1000 for each user to compensate for this. But then Eve could pay to herself in increments of 0.1 cent, which would lead to the potential gain of $50,000 for Eve and $10,000 in gas fees for people to stop her. So in general, it seems that the deposit should be infinite.