Extinction seems improbable.
AI will most likely loose interest in humans and allow a significant number of breeding pairs to exist indefinitely in a more or less natural state in the tropics.
Human IQ has been rising by about 3 points per decade, while computers have gone from 0 to 100 in less than 50 years. At the current rate of increase for AI, humans will be to AI as animals are to humans within a few decades.
Oh. You believe in that ideology.