To clarify: I do appear to be able to control the current to an extent.
When I do what the user manual says will turn it on, I can measure a 130uA current to ground on the outputs. When I do what the user manual says will turn it off, I can measure a 12uA current to ground on the outputs.
This is significantly less current, but it is still more than nothing and more than enough to produce a voltage reading.