I will describe joint work with Grant Barkley and Thomas Lam in which we study the Combinatorial Invariance Conjecture (CIC), which asserts that Kazhdan-Lusztig polynomials depend only on the combinatorics of Bruhat order. Motivated by the cluster structure on Richardson varieties, we prove the combinatorial invariance of the coefficient of q in KL polynomials for arbitrary Coxeter groups. We also prove the Gabber-Joseph conjecture for the second-highest Ext group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group.