According to D. Griffiths in Introduction to Electrodynamics, the second uniqueness theorem says
in a volume $V$ surrounded by conductors and containing a specified charge density rho, the electric field is uniquely determined if the total charge on each conductor is given.
Is it correct to think of this as a Neumann problem?
Recall that in a Neumann problem knowledge of the directional derivative of the unknown function on the boundaries is required to solve the problem uniquely. However, the second uniqueness theorem says that knowledge of the total charge on each conductor is sufficient. Now, the total charge on a conductor is not simply the directional derivative of the potential, but rather the surface integral of the directional derivative of the potential over the surface of the conductor.