I wouldn’t say they’re relying on it for operations. It’s not like they doing some internet controlled submersible, it is still watercraft with someone at the helm and more than likely, they use some type of local communication for the surface crew to communicate with the expedition crew.
Likewise for me. However, I'm on land, where the Starlink satellites don't have to relay comms between each other. It's a completely different story out in the middle of the ocean. Starlink only recently started their laser-based relay system, and I have not heard reports about its reliability. It might work great, I don't know. I do know that it was a difficult engineering problem to solve.
I'm speculating, but I'm fairly sure that they already have any and every satellite communication system available on-board, just as a back-up. If they're using Star link, it should be because it's working best.