Characterization of fixed points by the weak regularity inequality

Let $(M,d)$ be a spherically complete ultrametric space and let $F:M\rightarrow M$ be nonexpansive. For $x\in M$ with $x\neq F(x)$, consider the inequality

$$\liminf_{n\rightarrow\infty}d(F^n(x),F^{n+1}(x))<d(x,F(x)).$$

Fixed-point characterization. The following conditions are equivalent: (i) $F$ has a fixed point in every nonempty spherically complete subspace of $M$; and (ii) the displayed inequality holds whenever $x\in M$ and $x\neq F(x)$.

This gives an equivalence between a fixed-point property on all nonempty spherically complete subspaces and the stated orbit condition. The supplied text does not indicate whether this result is proved or remains open.