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That would be great if CD8+ TRegs did not have CCR5. The few papers on CD8+ TRegs I've seen indicate they do.

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Some do but not always. Migration of CD8+ T-cells is also not CCR5 dependent. In inflammation macrophages are far more proliferative than CD8+ T-cells so CD8+ T-cells add little to the inflammation. In anti-inflammation non-CD8+ Tregs are a much larger population then CD8+ Tregs and more likely to be CCR5 dependent. With CD8+ T-cells having such a small population how are they effective virus fighters? The volume of virus cells compared to the totality of body cells is quite small.