A note on combinatorial invariance of Kazhdan–Lusztig polynomials

We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the combinatorial invariance conjecture for the Kazhdan–Lusztig polynomials of the symmetric group. This conjecture has the advantage of being combinatorial in nature. The appendix by Barkley and Gaetz discusses the related notion of double hypercubes and proves an analogous conjecture for these in the case of co-elementary intervals.