2004 Mercury Grand Marquis LCM issue

When you say back running lights do you mean the tail lights? There are also reverse lights and brake lights. Are those functioning properly?
 
https://www.fixya.com/cars/t25752558-all_lights_work_but_not_tail_lights

Yes it's the running lights in the back only.. both are out.. (not the reverse lights, not the headlights, not the brake lights.. they all work including the blinkers work and the hazard lights work.. the map lights work..) the mechanic at the shop messed around with the LCM and they (the back running lights) worked; but said they the back running lights would probably stop working again and they did stop working again by the next day...
But when filling out the work request on your website it doesn't include indicating the backlights in the four options you can check.. I just need the service described in the video saying all four of the relays in the LCM are replaced.. should I just click the one that says "Vehicle external lights stay on all the time"? ..they don't stay on all the time but they stay off all the time.

We do replace all the relays that's correct. I've revised the wording on that listing so it should be up tomorrow. There are 4 relays. One for parking lights, one for headlights, one for interior lights, and the last one is a signal relay. If that is indeed the issue rest assured we have full bench-testing here to test for that functionality.

parking lights: https://www.propark.com/2021/12/17/what-are-parking-lights/
tail light: https://www.merriam-webster.com/dictionary/taillight
parking lights are not tail lights as described in the above reference.
You have added to the list of problems but there is no box to check for 'tail lights' nor 'back running lights'
These two lights in the backstay are on when the driver turns on the headlights.. they are always illuminated when the lights are on and
are red. We call them "taillights.' Which box do you think I should check?
Thank you for your assistance,

You are correct. For peace of mind, we can refurbish the LCM for you by replacing all 4 relays. If you're still having issues then you can be rest assured that it's not the LCM at fault. I didn't add tail lights because there isn't a dedicated relay for it as I mentioned. However, there is a signal relay that possibly could affect that. Wiring and/or switch is more likely at fault. However if your LCM hasn't been rebuilt yet by now, I highly recommend getting that done to eliminate at least one part of the lighting system.

can you estimate the turnaround time for the repair of LCM?

It'll take up to 3-5 business days however usually these can be quicker than that.