The idea that "The Battle of Axanar" was fought as part of a "Four Year War" was taken directly from a TOS Sourcebook supplement created by FASA for their "Star Trek Role-Playing" game circa 1984.That shouldn't be much of a factor in whether DSC will use Axanar as a plot element, really. It's not as if there'd be any demand for petty revenge, say, by writing a story that completely contradicts or shamelessly copies Peters'. Or for timidity, say, by steering clear of Axanar. Peters is too small a bug to be stamped on twice.
But we don't know of a "battle of Axanar" in onscreen terms. There's a "victory" associated with it, and a "peace mission", but not a "battle" as such. And no hint of Klingons, and only indirect suggestion that it would be around the time of DSC (in direct reading of the evidence, the non-peaceful part and the "victory" need to be while Kirk is still a cadet or even earlier on, and that's cutting it awfully close).
A Klingon war that takes one season sounds fine as such. But we need those 70 years of unremitting hostility, supposedly meaning the Klingons stayed quiet until the 2220s and only thereafter began what Spock counts as hostile action. How to mark the line? If DSC introduces the idea of war vs. peace, but the peace around the war still counts as unremitting hostility, then what were the relations like before the 2220s? Pretty flowers and honest handshakes?
Timo Saloniemi
https://www.scribd.com/doc/80485378/Star-Trek-Source-Book-the-Four-Years-War
^^^
So yeah, EVEN THAT wasn't an original idea from Alec Peters - he 'stole' that idea too.

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