Nobody has to. If you are contending that it dropped MORE than 2°, YOU have to demonstrate it. You're the one making the bold claim, here.
I warn you, it's not an obvious point. Ocean temperatures in specific locations can be dominated by the warm and cold ocean currents, rather than local weather conditions.
But let's assume that you have managed to show that the reefs died out in the last ice age. Maybe you've done the simple thing, and performed some carbon dating on the reefs themselves to show that they were quiescent during the last ice age, instead of doing complex numerical modelling of the ocean currents during that time period. Now you also have to show that the reefs could not have been recolonized from elsewhere, afterwards.
Can do?