Pointwise character bound extension to spherical functions on compact symmetric spaces

Let $X$ be an irreducible symmetric space of compact type, with spectral parameter $\mu\in \Lambda^+\subset i\mathfrak{t}^*$ and spatial parameter $H\in\mathfrak{t}$. Let $\psi(\mu,H)$ be the spherical function normalized by $\psi(\mu,0)=1$. Let $W$ be the Weyl group, and let $\Phi^+$ be the positive restricted root system, with root multiplicities $m(\alpha)\in\mathbb{Z}_{\geq 1}$. Spherical-function pointwise bound. There exists a uniform constant $C>0$ such that

$$|\psi(\mu, H)|\leq C\sum_{s\in W}\prod_{\alpha\in\Phi^+}\left(1+ \left|\langle s\mu,\alpha\rangle\sin\left(i\alpha( H)\right)\right|\right)^{-\frac{m(\alpha)}{2}}.$$

The inequality is proposed as a natural extension of the pointwise character bound for $\mathrm{SU}(3)$ to spherical functions on compact symmetric spaces. Its resolution is not specified in the supplied text.