Positive-characteristic $p$-adic interpolation conjecture for analytic automorphisms

Let $K:=\overline{{\mathbb F}_p}((t))$ and $K^{\circ}=\overline{{\mathbb F}_p}[[t]]$ be its valuation ring. Let $f:(K^{\circ})^r\to (K^{\circ})^r$ be an analytic automorphism satisfying $f=\operatorname{id}\mod t$. If there is no $n\geq 1$ such that $f^n=\operatorname{id}$, then the $f$-periodic points are not dense in $(K^{\circ})^r$ with respect to the $t$-adic topology.

Positive-characteristic $p$-adic interpolation conjecture. Under these hypotheses, the $f$-periodic points are not dense in $(K^{\circ})^r$ with respect to the $t$-adic topology.

This conjecture is proposed as a positive-characteristic substitute for consequences of the $p$-adic interpolation lemma, which fails in positive characteristic. Its status is not resolved in the supplied text.