Atolls do sink
It was actually Charles Darwin who first worked out how coral atolls form around a volcano which starts sinking back into the earth once it becomes extinct. From its shape, you can deduce that this one is indeed in the last stages of sinking beneath the ocean. Most of what was once a circular reef has already sunk beneath the waves.
How much of what they are seeing is sea-level rsing and how much is rock-level sinking is probably unanswerable. Regardless, the islander's plight is very real!