Is $\mathcal{H}_\delta^\alpha$ a metric outer measure?

Question

When going through a proof that $\mathcal{H}^\alpha:=\lim_{\delta\rightarrow 0}\mathcal{H}^\alpha_\delta$ is a $metric$ outer measure I found myself wondering whether for each fixed $\alpha$ and $\delta$, $\mathcal{H}^\alpha_\delta$ is also a metric outer measure. I haven't been able to prove it or find a counterexample yet. I want to believe that it's false, so I would appreciate it if someone could point an easy counterexample or, in the case that it's truly a source of proof. Here $\mathcal{H}^\alpha_\delta(A):=inf\{\sum_{j=1}^\infty (r_j)^\alpha\ |\ A\subset\bigcup_{j=1}^\infty B_{r_j}(z_j),\ r_j\leq\delta\ \ \ \forall j \}$ for any $A \subset\mathbb{R}^n$ And $\mathcal{H}^\alpha$ is, as it's usually called, the $\alpha$-dinensional spherical Hausdorff outer measure.