Our route was as follows:
Bullhead Trail > Boulevard Trail (with a 2.7 mile segment on the A.T. to get to Newfound Gap) > Alum Cave Trail > Rainbow Falls Trail > Trillium Gap Trail > Brushy Mountain Trail
My Gaia track did not start until we started hiking on the first trail (the first 0.4 miles are not on one of the 6 trails of the Le Conte Challenge. However, my Garmin track reflects our hike along this 0.4 miles, and that is the more accurate track to use for an FKT purpose. Also potentially worth noting is that I did not stop my Garmin track the exact moment I finished the hike (I forgot until a couple of minutes later!). By my hiking partner's timing we finished in 16:13, but I completely understand if the Garmin track time is the one you need to go with for the FKT.
And finally, because this route involves hiking trails that start from different places in the park, it is considered an acceptable practice to drive between trailheads to link them, which is what we did (between the A.T. on Newfound Gap and the Alum Cave Trailhead).
The only record I know of Tour Le Conte completions is on one of the websites I linked below, so that is where I gleaned my information to apply for this FKT.