J1 Visa beyond 5 years?

[WildeThing]

It is my understanding that all doctoral students sponsored for a J1 visa in the US do so as research scholars, which means they have a maximum stay of 5 years. While that theoretically aligns with most doctoral programs, many of them require more time to be completed (and some offer funding packages for 6 years already). Does anyone what happens if you need to extend your stay (and the program is willing to extend) but you have reached this limit? 

Some possibilities I have considered (with absolutely no knowledge to back this up):

- Apply for an extension (apparently there are some circumstances in which they are granted but I don’t know if they apply in this case)

- Get an Academic Training extension?

- Apply for an F1 in the final year? (Assuming it is possible, and would be shitty for eliminating J2 work potential)

- Finish the program long-distance? (I know some students can write their dissertations away from campus sometimes)

- Terminate the PhD (I guess least likely? But hey, who knows?)

Also, is it possible that a program that offers a 6 year package would refuse to sponsor a J1 because of this issue? 

Edited January 6, 2019 by WildeThing
Added a possibility

[marvel2375]

I would get in touch with the respective International Student Offices at the universities that you are applying to.

[WildeThing]

I definitely will, though my worry is that they will push me towards getting an F1 instead, and that is not option for me since I am coming with my wife who can't not work for 6+ years (but I don't know if that's a valid reason (in their eyes) to need a J1 and maneuver the additional bureaucracy).