Dynamical nonperiodicity for automorphisms with small spectral radius

Let ${\mathbf{k}}$ be an algebraically closed field of characteristic $0$, let $X=\mathcal{M}(A)$ be the affinoid space under consideration, and let $f:X\to X$ be an automorphism with $\rho(\Delta_f)<1$. Small-spectral-radius nonperiodicity conjecture. If $f$ is not of finite order, then there is a non-empty open subset $U$ of $X$ such that every ${\mathbf{k}}$-point $x$ in $U$ has infinite orbit. The statement is presented as a more general form of a cited conjecture and is motivated by consequences of the $p$-adic interpolation lemma; its resolution is not given in the source.