Extension of Theorem 1 to arbitrary spherically complete sets

Let $A$ and $B$ be nonempty spherically complete subsets of an ultrametric space, and let Statements (i)--(iv) denote the four statements from Theorem 1. Extension conjecture. Statements (i)--(iv) remain valid for all such sets $A$ and $B$, even when

$$\delta(B)\leq \operatorname{dist}(A,B)$$

does not hold.

The claim proposes removing the inequality hypothesis from Theorem 1 while retaining all four conclusions. The supplied text does not indicate whether this extension has been proved or disproved.