Towards a Proof of the 2-to-1 Games Conjecture?

We propose a combinatorial hypothesis regarding a subspace vs. subspace agreement test, and prove that if correct it leads to a proof of the 2-to-1 Games Conjecture, albeit with imperfect completeness. This paper presents the second installment in a line of work by various subsets of the authors (with additional contributions by Barak, Kothari, and Steurer (ITCS’19)), which led to a proof of the 2-to-2 Games Conjecture.