I got one of those on second day of owning a Taycan, at a start of a 3,500 mile trip home from the dealer (it was mid winter, hence winter tires).
Noticed it via the tire pressure display (car did not warn yet, but I noticed one tire was lower than the others). Plugged in at a nearby 24hr tire place (it was past midnight). Lasted the entire trip and rest of winter without issues. Talked to a tire store about patching it over the plug, they said not to worry about, the plug should last the life of the tire just fine.
I've since picked up a tire plug kit similar to what @enzo5000 mentioned in post #8. Should this happen again to me in the middle of a trip, I can patch it myself.