Which driving school? NASA? BMWCCA? PCA? Other? I don't know of any marked differences between the clubs re tech inspection, but someone else might. I do know that BMWCCA requires you to have changed your brake fluid within the past 90 days. Not sure how that particularity might translate over into tires.
I'm not sure what the actual depth number might be, but they will be looking for any chunking or exposed cord, especially on the outer corners of the tires, where you will see the most wear from HPDE. For example, my street tires, which I tracked previously but don't anymore due to a new setup, have visible wear on the corners, but would pass inspection because they have tons of tread on the flat and the corners show no chunking etc. However, a friend's front tires started chunking on the corners. They were very new tires but the track days had taken their toll on the corners, and those tires would never have passed inspection. In fact, she had to quit early the day the chunking happened. It was not safe to continue.
If you have a real question, I'd recommend playing it safe and get new tires. If you plan on doing more tracking on the next tires, you might do some shopping for the model of tire you might prefer. Recommend you take this question to the Tires forum here, and post some photos of the tires you have.