The problem, as it's been reported, is that the "control input" of the antenna
(I assume you have a separate +12V feed and ground (or grounded to chassis) to
it?) is very high impedance. As the empeg essentially disconnects its end of the
wire on "turn-off", the "charge" on the control input doesn't "bleed away" to ground.
The resistor offers a path to "bleed out" through (aka pull-down)

You can apply the 1kOhm resistor just about anywhere as long as the resistor
goes between the signal line and ground, for instance at the plug that connects to
the tuner module. It could also be applied internally, inside the module.

IMO, since this fix is more or less antenna specific, I'd look at placing the resistor by the antenna.

HTH,

/Michael

Edit: A little ascii art to assist

Tuner  -blue------------------*---------------------------*-----------Antenna-- +12V

module                                                    |                |

                                                          |_resistor_______*____Ground

The blue cable comes out of the tuner module plug, is crimped or soldered to the wire that controls
the antenna. The resistor goes between this line (anywhere, but I'd do it by the antenna,
if at all accessible) and ground (anywhere).
* denotes connections.