Scientific American is sadly no longer the organization they used to be. Bad headline, it didn't cite previous work perfectly but still provided a novel result. Here is one of the authors of a paper it built on, talking about the result

https://mathoverflow.net/questions/513866/what-are-the-key-new-ideas-in-the-proof-of-nonsoficity-of-groups-in-openai-s-con

So, that is the proof of Proposition 2.3. It is a creative and at the same time elementary construction that uses not just property (T) for an application of my result with Kun, but also for the ambient group G in order to overcome the problem, that the Γ-components might be of different size. Once this is achieved, the rest of the argument is straightforward.

I think it is too early to judge if this construction will serve any other purpose than providing the right framework to apply these earlier results efficiently. On the other side, I was looking myself for such a mechanism ever since we wrote the paper in 2019 and admire the efficiency of this construction.