Flip Combinatorial Invariance and Weyl groups

In this work, we investigate the approach via flipclasses to the Combinatorial Invariance Conjecture for Kazhdan–Lusztig polynomials of all Coxeter groups. We prove the combinatorial invariance of Kazhdan–Lusztig $\tilde{R}$-polynomials of Weyl groups modulo $q^7$ and of Kazhdan–Lusztig $\tilde{R}$-polynomials of type $A$ Weyl groups modulo $q^8$. As a consequence, the Combinatorial Invariance Conjecture holds for all intervals up to length 8 in Weyl groups and up to length 10 in type $A$ Weyl groups.