Bridge positions of links that cannot be monotonically simplified

For any pair of integers $m$ and $n$ such that $3 < m < n$, we provide an infinite family of links where each link in the family has a locally minimal $n$-bridge position and a globally minimal $m$-bridge position. We accomplish this by applying the criterion of Takao et al. The $n$-bridge position is interesting because the corresponding bridge sphere is unperturbed, so it must be perturbed at least once before it can be de-perturbed to attain a globally minimal $m$-bridge sphere.