If you pay someone more than $0, that’s when you need to pay them at least minimum wage.
> FLSA prohibits public sector employees from volunteering, without compensation, to work additional time performing the same type of services or work for which they are employed. For example, a police officer can properly be considered a volunteer, and need not be paid, for reading to school children at the municipal library. The same officer must be paid, however, and cannot be considered a volunteer, when directing traffic for a city-sponsored road race in the same municipality where the police officer is employed.
So I guess if you already have a position in another state school, you apparently can't take this one.
[1] https://www.wardandsmith.com/articles/wage-hour-law-voluntee...
This position is unpaid at a public not private employer. In this case the commonly understood meaning of "volunteer" is likely the same as the legal one.
The "without salary basis" looks pretty clear to me, and I'm sure someone holding a PhD could spot that as well.