Suppose you place a number of rotating black holes in linear sequence (rotating around the same axis) between two stars at distance d (assume as tightly packed as practical for purposes of calculation). If a ship enters each ergosphere and goes out into the next one, can it make the roundtrip back to home in less time than 2d/c (home local time)?

What would be the simplest calculation to see that it cannot?

Edit to make the ergosphere overlapping region more symmetric, imagine there is a symmetric sequence of black holes in front of the treadmill arranged like this:

A kerr blackhole treadmill with an overlapping region of ergospheres.

The symmetry in this arrangement should cancel the angular components (at least in a small region in the middle of overlapping region). Obviously the geometry for this thing is highly unstable.