Combinatorial invariance for the coefficient of $q$ in Kazhdan-Lusztig polynomials

Abstract: We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan-Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture for Bruhat intervals of length at most $6$. We also prove the Gabber-Joseph conjecture for the second-highest $\mathrm{Ext}$ group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group.