Under what circumstances is the antenna supposed to be up? As far as
I can tell, it is up whenever the empeg display is on, although I'm pretty
sure I have the tuner flying lead connected to the same place the original
head unit used to control the antenna. I have tried turning off
all the radio-related functions I can see but the antenna stays up.

Is there a way to see what state the player has commanded?

Is there a sequence I can send to the tuner module to switch the output
off and on?

I am aware of the FAQ's suggested resistor-to-ground fix but haven't had
a chance to try it yet. I'm a bit doubtful this will be the solution
as my voltmeter shows a clear change of state from 0 to 12V and back
to 0 correlated with the display state, so it doesn't seem too likely
anything is floating, but I'll give it a try anyway.