Metric uniform simplicity of simple metric ultraproducts

Let $({\mathcal G}_{\mathrm{met}},\|\cdot\|)$ be a metric ultraproduct of metric groups. A metric group is metrically uniformly simple if, for every $r>0$, there is $N\in\mathbb{N}$ such that $C_N(g,G)=G$ for all $g\in G$ with $\|g\|>r$, where $C_N(g,G):=(g^G\cup g^{-1G})^{\leq N}$. The metric uniform simplicity conjecture. If $({\mathcal G}_{\mathrm{met}},\|\cdot\|)$ is simple, then ${\mathcal G}_{\mathrm{met}}$ must be metrically uniformly simple. The paper identifies this as a conjecture suggested by the fact that all known simple metric ultraproducts have the stronger property, while the available results establish only bounded simplicity.