In the cyclic case Chen–Chen [11, Theorem 1.1] have a worse decay $O(r^{-2})$ of an ALF metric of type $A_{k}$ to $g_{k+1}$. However, from the explicit description of cyclic ALF gravitational instantons as multi-Taub–NUT spaces, as we will recall below, it is clear that one can always change coordinates by a translation on $\mathbb{R}^{3}$ so that every $A_{k}$ ALF space satisfies the stronger decay stated in Definition 3.6.(i).