2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/223563We explain a strategy for a proof of the positivity of all coefficients of Kazhdan-Lusztig-polynomials for arbitrary Coxeter groups by constructing spaces whose dimensions we conjecture to be these coefficients.Version accepted to be published, with corrected proofs and enhanced resultsRepresentation TheoryCombinatorics20F55, 17B10text