Burrull, G.; Libedinsky, N. & Plaza, D. (2021), 'Combinatorial invariance conjecture for $A_2$' , cite arxiv:2105.04609Comment: 21 pages, 9 colored figures. The new Preliminaries section includes the new Proposition 2.5 which simplifies the proof of the Main Theorem. New background material for the affine Weyl group was included. The old Appendix was replaced by the proofs of Propositions 3.1 and 3.3 in greater detail. The acknowledgments section was added. Final version .