Kartik Prabhu
Finally, managed to do something with twistors https://arxiv.org/abs/2111.00478 ! 17 year-old self would be very happy :D.
Turns out you cannot impose the twistor equation at null infinity. But, you can impose three of the six components of it! You get an infinite-dimensional space of spinor solutions, BMS twistors.
Then, you can take two of these BMS twistors and generate a complex BMS symmetry at null infinity!
Not sure why this works but seems pretty neat.