Not automatically, but the Mint DVM would just have all the mint actions to manage mints. The mint state would then be stored as an encrypted Nostr event in a relay. The DVM is stateless.
Discussion
The nice thing is you don't need to hide the state from anyone, an ecash mint can be fully observable except for its private keys. In fact, a db accessible by anyone can be extremely useful for ecash.
In fact, if the db itself isn't managed by the mint, it would lower the trust requirements in the mint (because public access to it allows you to prove fraud more easily).
You can even have a history if Mint calls if you don't delete the requests and responses to the DVM. Full audit.
Yes, fraud proofs work similar to that: you need to publicly prove that you requested a transaction but didn't get the intended response. If these requests (or the mint's commitment to process a valid request) could be audited by anyone, it should be possible to prove a mint is denying it's service or acting maliciously.
Fraud proofs are hard because you can’t easily prove that you aren’t hiding shit from the mint. Broadcast it on a relay the mint can’t read from (IP blacklist for example), and you just framed them for something they couldn’t respond to.
That's why it's best to combine it with a challenge-response scheme: To prove that the mint has replied correctly to a request, it just replies correctly to it.