Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006, as well as Joyal's own account in The Theory of Quasi-Categories and its Applications from 2008.
I have seen unsourced claims that this model structure may originate in 1980s, possibly in the correspondence between Joyal and Grothendieck. However, an examination of the only available letter from Joyal to Grothendieck does not reveal any material on quasicategories. I was unable to find any other correspondence between Joyal and Grothendieck.
In his 2008 notes cited above, Joyal says “The results presented here are the fruits of a long term research project which began around thirty years ago.”, which appears to indirectly corroborate the above unsourced claims.
When did the Joyal model structure on simplicial sets originate? Is there any written source that confirms its existence in 1980s?