On Cowling's $L^p$-integrability conjecture for Kunze--Stein groups

We prove a stronger form of Cowling's $L^p$-integrability conjecture for an admissible class of Kunze--Stein groups containing all $S$-algebraic groups and all closed CCR boundary-transitive subgroups of tree automorphism groups. We also disprove a cyclic version formulated by Samei--Wiersma.