Reject the cookie, because
SameSite=None requires SecureModern browsers reject the cookie, because a SameSite=None cookie must also have Secure. Without Secure, the browser blocks it. So "Store it and send it on all cross-site requests" is wrong: the browser does not store it at all. It also does not quietly change it to SameSite=Strict.
The fix is SameSite=None; Secure, served over HTTPS. Even then, third-party cookie restrictions in the browser may still block the cookie.
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