I don't understand why anyone would use passwordless disk encryption. It just seems inherently vulnerable, especially with the threat model of physical compromise.
Entering a password on boot isn't even that much work
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Entering a password on boot isn't even that much work
There is precisely zero chance that the relevant IT security goons would allow any kind of remote KVM/LTE connection.
Any change the untrusted local staff could make to the server, they could also make to the KVM machine (e.g. turn it into a keylogger).
Now you have the same problem but with a smaller computer.
You cannot turn untrusted systems into trusted systems by adding more untrusted systems.