The upper-bound conjecture for bounds of bounded affine Hecke algebra representations

Let $(\pi_{J,\mathsf{v}},M_{J,\mathsf{v}},\mathsf{B}_{J,\mathsf{v}})$ be a bounded representation, let $\mathbf{a}_{J,\mathsf{v}}$ denote its bound, let $J$ be the relevant subset of simple reflections, and let $L(\mathsf{w}_0)$ be the weighted length of the longest element of the finite Weyl group. Upper-bound conjecture. The bound satisfies

$$\mathbf{a}_{J,\mathsf{v}}\leq L(\mathsf{w}_0),$$

with equality if and only if $J=\emptyset$. This conjecture generalizes the upper bound proved for the simplest weighted $J$-parameter system; the precise bound is expected to be connected with Lusztig's $\mathbf{a}$-function, Macdonald's $c$-function, and Opdam's Plancherel theorem, but the conjecture is not resolved in the supplied text.