Answer
Before troubleshooting SSO or authentication further, check whether that specific projection has simply had its "Int" Compatibility Application Type deliberately disabled, leaving only "Main" enabled — a developer can intentionally restrict a projection to a single application type, and this produces a login-prompt-that-never-accepts-anything symptom that looks exactly like an SSO/authentication problem but is actually the application-type restriction working as designed; check the projection's Compatibility Application Type settings before assuming it's an SSO configuration issue.