Absolute convergence of the geometric conjugacy-class expansion

Let $G$ be the reductive group, let $f\in C_c^\infty(G(\mathbb A)^1)$, and for each geometric conjugacy class $C$ in $G$ let

$$K_{P,C}(x,y)=\sum_{\substack{D\subset P/N\operatorname{Ind}_P^G D=C}}\sum_{\gamma\in D}\int_{\mathbf N} f(x^{-1}\gamma n y)\,dn$$

be its contribution to the kernel function, with

$$J_C^T(f)=\int_{G\backslash\mathbf G^1}\sum_P K_{P,C}(x,x)\hat\tau_P^T(x)\,dx.$$

Absolute convergence conjecture. For every $f\in C_c^\infty(G(\mathbb A)^1)$, one has

$$\sum_C\int_{G\backslash\mathbf G^1}\left|\sum_P K_{P,C}(x,x)\hat\tau_P^T(x)\right|\,dx<\infty.$$

This condition would justify the expansion of the trace distribution into contributions from geometric conjugacy classes, namely $J^T(f)=\sum_C J_C^T(f)$. The source presents it as the condition needed for convergence and does not provide evidence of a resolution here.