Based on the ending of tadc and things Goose has said about Jax (namely: she's sad people stopped torturing him, she relates to him most of anyone on the cast), i can really only see him as an embodiment of a "what if there was a version of myself that never got better, that person would suck i think". And man, do i completely understand that situation... I made an OC of my own with that exact idea in mind and never really stopped torturing him, though I've gotten a lot nicer to him over the years.
I don't think he'd ever transition, even if he got un-abstracted. I do not think there is or could have ever been a good end for Jax, because his character was to be a cautionary tale for not accepting oneself, as well as a "what if" for Leeroy Mateo not getting better, just like I said above.