Let $G$ be a finitely generated group, with a fixed word metric $d$ coming from some finite generating set. Fix $n\in \mathbb{N}$, and suppose that for all finite $S\subseteq G$ with $\mathrm{diam}(S)=D>n$ there exists $g\in G$ such that $d(g,s)\le D-1$ for all $s\in S$ and $d(g,s_0)\le 1$ for some $s_0\in S$. Thinking in terms of the Cayley graph, this means you can walk from $s_0$ to any other place in $S$, along a path whose length is at most the diameter of $S$, specifically passing through $g$ as the first step.
One example where this happens is if $(G,d)$ has the Helly property, meaning any collection of balls that pairwise intersect in fact have a common intersection, see https://arxiv.org/abs/2002.06895. Indeed, since $S$ has diameter $D$, we know that the balls of radius $D-1$ centered at the points of $S$ together with a single ball of radius $1$ centered at some (any) $s_0\in S$ pairwise intersect, hence all intersect, and now just take $g$ to be anything in this intersection.
But the Helly property feels like overkill to get this property to hold. First of all, we only applied it to a very specific collection of balls. Secondly, it worked for any choice of $s_0\in S$, whereas in the property I want, it isn't required to work for every choice of $s_0$, just some. Also, I only need this to happen for sufficiently large $D$. (But maybe for groups the only way to get all this stuff is via the Helly property, who knows.)
Question: Are there any groups that satisfy this property without having the Helly property?
The reason I'm interested in this is, I think I can prove that any group with this property admits a contractible Vietoris-Rips complex, which is a reasonably strong thing. But this is already known for groups with this Helly property, so I can't tell whether anything actually new emerges from this.
