The cell-recognition conjecture for bounded affine Hecke algebra representations

Let $\Gamma_{J,\mathsf{v}}$ be the cell recognised by the bounded representation $(\pi_{J,\mathsf{v}},M_{J,\mathsf{v}},\mathsf{B}_{J,\mathsf{v}})$, let $\mathbf{a}_{J,\mathsf{v}}$ be its bound, let $\mathbf{a}$ be Lusztig's $\mathbf{a}$-function, and let $(W,L)$ be the weighted Coxeter group. Cell-recognition conjecture. The following hold:

1. $\mathbf{a}(w)=\mathbf{a}_{J,\mathsf{v}}$ for all $w\in\Gamma_{J,\mathsf{v}}\cap W$.
2. The set $\Gamma_{J,\mathsf{v}}\cap W$ is contained in a two-sided Kazhdan–Lusztig cell of $(W,L)$.

The conjecture links bounded representations with Kazhdan–Lusztig cells and, together with the product formula conjecture, would give a conjectural formula for Lusztig's $\mathbf{a}$-function on recognised elements. It remains open in the supplied text.