Should we be worried about Jon snow's title in the North?

On several occasions the Northerners have declared that the North will know no king whose name is not Stark.

Following the revelation at the season six finale, that L + R = J [Lyanna Stark + Rhaeghar Targaryen = Jon Snow] which means Jon is not a Stark but a Targaryen, what do you think would happen?

ALSO READ: Winter has finally come! [Major spoiler alert]

Would Jon Snow be stripped of all the his titles in the North? Or would be allowed to reign as the king in the North.

Drop your comments in the comment section below, Winter is coming!