Completely compressible Bruhat intervals and Kazhdan–Lusztig polynomials

Our main result is that the recently proved combinatorial invariance property for Kazhdan–Lusztig polynomials on lower Bruhat intervals still holds for Bruhat intervals whose top element is critical. We conjecture that our theory also extends to completely compressible Bruhat intervals in type A,D,EA,D,E, and we have checked this conjecture up to types A8,D7,E6A8,D7,E6 with a computer. Positive results in related special cases are also presented.