Sharp estimate for the Dunkl kernel of type $A_n$ via shortest Weyl-chamber realizations

Let $A_n$ be a root system with positive roots $\Sigma^+$, Weyl group $W$, and positive Weyl chamber $C^+$. For a Weyl chamber $C$, let $\mathcal{Short}(C)\subset W$ be the set of shortest realizations of $C$: if $wC^+=C$ and $w=\sigma_{\alpha_1}\cdots\sigma_{\alpha_s}$, then a realization $(\sigma_{\alpha_1},\ldots,\sigma_{\alpha_s})$ is shortest when $s$ is minimal. For $\lambda\in C^+$ and $X\in C$, write $X^+$ for the corresponding point in the positive chamber. Sharp-estimate conjecture. There exists $S\in\mathcal{Short}(C)$ such that

$$E_k(X,\lambda)\asymp e^{\langle\lambda,X^+\rangle}F(X,\lambda),$$

where

$$F(X,\lambda)=\frac{1}{\prod_{\alpha>0}(1+\alpha_\lambda\alpha_{X^+})^{p(\alpha)}},$$

with $p(\alpha)=k+1$ if $\sigma_\alpha\in S$ and $p(\alpha)=k$ otherwise. In particular, for $X\in C^+$,

$$F(X,\lambda)=\frac{1}{\prod_{\alpha>0}(1+\alpha_\lambda\alpha_X)^k},$$

and, for $X\in\sigma_\beta C^+$,

$$F(X,\lambda)=\frac{1}{(1+\beta_\lambda\beta_X)^{k+1}\prod_{\alpha\in\Sigma^+\setminus\{\beta\}}(1+\alpha_\lambda\alpha_X)^k}.$$

The preceding type $A_n$ theorem and estimates for the type $A_1$ Dunkl kernel support this proposed general estimate; the source gives no resolution.