The product formula conjecture for bounds of bounded affine Hecke algebra representations

Let $\Phi$ be the root system, let $(\pi_{J,\mathsf{v}},M_{J,\mathsf{v}},\mathsf{B}_{J,\mathsf{v}})$ be a bounded representation, and let $\mathbf{a}_{J,\mathsf{v}}$ denote its bound. For $\alpha\in\Phi$, define $\mathsf{q}_{\alpha}\in\mathbb{Z}[\mathsf{q},\mathsf{q}^{-1}]$ by

$$\mathsf{q}_{\alpha}=\mathsf{q}^{L(s_i)}\quad\text{if $\Phi$ is reduced and $\alpha\in W_0\alpha_i$},$$

and, if $\Phi$ is not reduced,

$$\mathsf{q}_{\alpha}=\begin{cases} \mathsf{q}^{L(s_i)}&\text{if $\alpha\in W_0\alpha_i$ with $i\neq n$},\\ \mathsf{q}^{L(s_n)-L(s_0)}&\text{if $\alpha\in W_0\alpha_n$},\\ \mathsf{q}^{L(s_0)}&\text{if $\alpha$ is long.} \end{cases}$$

Product formula conjecture. If the representation is bounded, then

$$\mathbf{a}_{J,\mathsf{v}}=L(\mathsf{w}_0)-\frac{1}{2}\deg\prod_{\alpha\in\Phi}'\frac{1-\mathsf{q}_{\alpha/2}^{-1}\mathsf{v}^{\alpha^{\vee}}}{1-\mathsf{q}_{\alpha/2}^{-1}\mathsf{q}_{\alpha}^{-2}\mathsf{v}^{\alpha^{\vee}}},$$

where $\prod'$ means that factors in the numerator or denominator that are $0$ are omitted. This is presented as a stronger conjecture than the upper-bound conjecture and is intended to give the precise value of the bound; its status is open in the supplied text.