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The aim of this work is to prove a conjecture related to the CombinatorialInvariance Conjecture of Kazhdan-Lusztig polynomials, in the parabolic setting,for lower intervals in every arbitrary Coxeter group. This result improves andgeneralizes, among other results, the main results of [Advances in Math. {202}(2006), 555-601], [Trans. Amer. Math. Soc. {368} (2016), no. 7, 5247--5269].

The combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials of lower intervals