Boundedness conjecture for relative Shalika germs

Let $A_e^{w_i}$ and $A_e^{w_{G_n}}$ be the domains associated with the relative Shalika germs $K_e^{w_i}$ and $K_e^{w_{G_n}}$, respectively, and let $\Delta$ be the discriminant function. The germs are indexed by $i=1,2,\ldots,2^n-3,2^n-2$. Relative Shalika-germ boundedness conjecture. For some $\delta>0$, the functions

$$\left\vert K_e^{w_i}\Delta^{\frac12-\delta}\right\vert$$

and

$$\left\vert K_e^{w_{G_n}}\Delta^{\frac12-\delta}\right\vert$$

are bounded on $A_e^{w_i}$ and $A_e^{w_{G_n}}$, respectively. This is presented as a reduction of the Bessel-integral conjecture; the source gives no resolution in general. A later, weaker-looking formulation isolates the longest-Weyl-element germ with a parameter $\delta_n>0$.