They "cheated" because they did the easy part, and the "other component" has no known solution.
What they're handwaving away is the real problem. The reason why current methods work for lower frequencies and not for higher frequencies is that your microphone, emitter, and ear all need to be enclosed within a volume << the minimum wavelength you can handle, to eliminate sound field and propagation time effects and let you treat the noise cancellation problem as a point problem in space.
Once you put the ears in open air and the emitters away from them, the problem isn't figuring out the sound that's going into the users' ears, it's figuring it out in advance. The ANC speakers have to send the complement signal before the original signal arrives at the ears, so they both arrive coincidentally.
In other words, the "missing part" to make this research work is time travel (if you use their sensing technology as the only input). Or maybe full-volume sound field sampling and characterization technology (if you don't). Neither of which exist, and which are orders of magnitude more difficult than this research.
The handwave part is "The constraint of the possible locations [of microphones] would affect the control performance and this remains a topic to be further studied in the ANC community."
Yes, it would completely destroy high frequency performance, which is what they were trying to achieve, in anything but lab scenarios (such as the sound sources still being loudspeakers and the microphones being directly in line between them and the user). Real world noise doesn't come from loudspeakers, it comes from all around you. Good luck using distant microphones to compute the expected sound at a user's ear in advance, with any kind of accuracy in the high frequencies.
At best I imagine they could achieve adaptive cancellation for a set of slowly moving pointlike sources in an otherwise simple room, with a number of microphones greater than the number of sources. It's like RF MIMO systems. But again, this is orders of magnitude more work to implement, and real world constraints are going to kill your high frequency response. And for larger sources - forget it. You just can't characterize the transform for that. Not enough dimensions in your input data to solve for it. So anything mechanical, stuff where the noise isn't coming from a literal speaker with a 1-dimensional input signal - nope. As the uncertainty and source size grows, your high frequency response goes down the drain.