There's no public WI-FI on the island, so the app won't do the messaging, and will only let you view what you've already downloaded to the app on the ship, like the activities for the day.
In my experience, they are both right, even on the same day.
When we first got off the ship, messaging worked fine. Then, on the tram, it did not, and I thought we'd have to go without it for the rest of the day, but when we reached the beach, it worked again. We could get messages back and forth all day.
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