Combinatorial invariance for Kazhdan-Lusztig $R$-polynomials of elementary intervals

We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig $R$-polynomials in the case of elementary intervals in $S_n$. This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.