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Hitchin maps emerge in the study of principal G-bundles on smooth projective curves. Oblomkov, Rasmussen, and Shende strikingly conjectured that for GL_n, the cohomology of their fibers bears a precise relation to combinatorial invariants of certain knots/links. We not only extend their conjecture to other G, but also upgrade it to a relation between two graded characters of the Weyl group of G, defined by contrasting interpretations of the associated braid group: one combinatorial, one topological. Under a condition we call homogeneity of mild defect, we relate the new conjecture to a known relationship between finite Hecke algebras and rational Cherednik algebras.