Just found this Press Release: Rural Connectivity Group Awarded Chatham Islands Contract – Rural Connectivity Group (thercg.co.nz)
The TL:DR is they are building 5 4G towers there that will be on Satellite backhaul.
Guess then it comes down to choice with how much a user wants to spend on a RCG connection or Starlink.
Watching starlink.sx it doesn't look like the Satellites need to talk to each over to reach the Chathams, the Hinds uplink site appears to reach satellites flying over the islands.