> [...] But if the immunity really is strain-specific
Reinfections have been recorded from different strains because it's much easier to demonstrate that someone has been reinfected with a new strain. If someone has been reinfected with the same strain then it is hard to distinguish that from an infection that wasn't fully cleared originally. It doesn't necessarily mean that immunity is strain-specific.
(I don't know why you were downvoted; your comment seems reasonable to me!)