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Everyone, including Classics scholars I know, when speaking English, would pronounce Ajax as in 1. above. That is even when Classics scholars are speaking with other Classics scholars. Likewise, I’ve never heard anyone, when speaking English, use the classical pronunciation of Caesar, or Achilles, or Jupiter, or Juno, even in academic circles. I’ve heard the pronunciation go back an forth when speaking English vs quoting a text in Latin or Greek.