Let's call a metric space $ X $ strongly contractible if there exists a function $ \rho : \mathbb{R}_+ \to \mathbb{R}_+ $ such that for every ball $ B(x;r) $ around a point $ x \in X $ we have:

• $ B(x;r) $ is contractible inside $ X $. So there is a map $ h : B(x;r) \times [0,1] \to X $ such that $ h(p,0) = p $ and $ h(p,1) = x $ for all $ p \in B(x;r) $.
• We can choose $ h $ to have the property $$ d(h(p,t), h(p',t)) \leq \rho (d(p,p')) $$ for all $ p,p' \in B(x;r) $ and $ t \in [0,1] $.

I want to know for which finitely presented groups $G$ is $EG$ (the universal cover of it's classifying space) strongly contractible? I understand that these spaces $EG$ are uniformly contractible, so all $R$-balls can be contracted within some $S$-ball, but I don't see why the second property would hold.

Any help or pointers will be greatly appreciated, I don't know where to start looking for relevant literature.