Existence of balanced cell representations for affine Hecke algebras

An affine Hecke algebra is an algebra associated with an affine Weyl group and a choice of parameters; a balanced system of cell representations is a system of representations satisfying the balance conditions used in the paper. Existence conjecture. For each affine Hecke algebra, there exists a balanced system of cell representations for each choice of parameters. Assuming this conjecture, Lusztig's $\mathbf{a}$-function satisfies $\mathbf{a}(w)\leq \mathbf{a}_{\Gamma}$ whenever $w\in\Gamma$, with equality when the system satisfies the extra axiom ${\bf B4}'$.