Cowling's conjecture on coefficient functions of Kunze–Stein groups

Let $G$ be a Kunze–Stein group, and let $\pi:G\to B(H_\pi)$ be a unitary representation admitting a cyclic vector $\xi\in H_\pi$. Write $\pi_{\xi,\xi}(s)=\left\langle\pi(s)\xi,\xi\right\rangle$ and let $A_\pi$ denote the associated coefficient space. Cowling's conjecture. If

$$\pi_{\xi,\xi}\in L^p(G)$$

for some $2<p<\infty$, then

$$A_\pi\subset L^p(G).$$

This is a 1978 conjecture of Cowling concerning coefficient functions on Kunze–Stein groups. The paper presents a near solution, but the supplied text does not state that the conjecture is completely resolved.