Automorphic matrix-coefficient bound for low-rank absolutely almost simple groups

Let $K$ be a global field, let $\mathbf G$ be a connected absolutely almost simple $K$-group, and let $W_f$ be a compact open subgroup of $\mathbf G(\bA_f)$. For $f,h\in L^2_{00}(\mathbf G(K)\backslash \mathbf G(\bA))$, assume that $f$ and $h$ are $U_\infty\times W_f$-invariant unit vectors. Automorphic bound. There is a constant $c_{W_f}>0$, depending only on $\mathbf G$ and $W_f$, such that

$$|\langle f,g.h\rangle|\leq c_{W_f}\cdot \xi_{\mathbf G}(g)$$

for all $g\in \mathbf G(\bA)$. This conjectural estimate would provide uniform decay of matrix coefficients for the automorphic spectrum in $L^2_{00}$, particularly when the $K$-rank is at most one, where the corresponding bound for arbitrary infinite-dimensional representations does not generally hold.