The general consensus amongst the staff at East Grinstead until recently, and I believe still is, that they believe that Uckfield Line seasons are valid there, so I'd hope you wouldn't have a problem.
However, I've never once met another member of rail staff who's ever seen that document (as found via the link above) when I've shown it to them, and it's a bit suspect in that it doesn't show any form of source or date of issue.
Local training always used to be pretty good on this particular issue, though. I'm not sure about the current training for new Southern staff.