Generalized convergence conjecture for invariant triple products

Let $G={\rm SO}(d,1)^0$, let $k_0={\underline k}(w_0n_0)$, and let $\rho$ denote the half-sum of the positive roots. The lemma establishes convergence of

$$\int_G {\underline a}(w_0n_0y)^\rho {\underline a}(w_0y)^\rho {\underline a}(y)^\rho\,dy.$$

Generalized convergence conjecture. The same convergence assertion should hold for any semisimple group $G$ with finite center and generic $n_0\in N$. This extension would generalize the convergence estimate needed to construct invariant distributions for triple products beyond the rank-one group considered in the lemma. The source states the assertion without providing a proof or resolution.