The Liturgy is attributed to him, or at least key elements are, especially the Anaphora. The claim is reasonable, so far as I know, i.e. there is no reason to doubt the attribution.

I don’t quite have the time and resources to follow this up right now (I'd have to do some inter-library loans with the Orthodox seminary in town, and given that it's December, I suspect I wouldn't get my hands on them until mid January), but I can set you on exactly the right path: Robert F. Taft. He is the author of a 6-volume History of the Liturgy of St. John Chrysostom, one of which I believe remains unpublished as yet (vol 3), as well as extensive articles on the history of Byzantine liturgy. If you can get access to some of his writings, there lies the answer (buried amidst a comprehensive history of the Liturgy).