Tax haven rules are on the US side, not the Cayman side. FATCA makes Cayman way more transparent than the US itself.
Like, what does "stop being a tax haven" even mean? "Start charging a corporate income tax you don't need"?
A lot of people who don't know anything about finance think that "offshore bank accounts" is a magical tax loophole. They don't know what "tax haven" really means, they just hear it repeated often.