Unitary $L^{p+}$-representations of almost automorphism groups

Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze-Stein property, and let $\pi$ be a unitary representation of $H$. We show that the representation $\tilde{\pi }$ of $G$ induced from $\pi$ is an $L^{p+}$-representation if and only if $\pi$ is an $L^{p+}$-representation. We deduce the following consequence for a large natural class of almost automorphism groups $G$ of trees: For every $p\in (2,\mathrm{\infty })$, the group $G$ has a unitary $L^{p+}$-representation that is not an $L^{q+}$-representation for any $q<p$. This in particular applies to the Neretin groups.