If they've recovered, they're immune. If you weren't immune, you couldn't recover - it's the same system.

The only question is how long-lasting it is, and so far the answer is "indefinite / a long time".

Other people have caught it and recovered, but it was not known whether that gave them immunity. What's different here? In fact, some people have caught it twice, so you can obviously recover without gaining immunity.

But if they suddenly start dying

“Huh well that was interesting”

so honestly theyll say whatever is convenient. nobody knows anything, nobody has any plan, we’ll see