So let's assume that the relatively ambiguous Tower of Joy scene in season 6 confirms that Jon is indeed
Lyanna Stark's child, sired by Rhaegar Targaryan.
In that case, considering the current situation in Westeros as of season 7 (or as of book 5 + released chapters of book 6), how does this get proven in the eyes of the lords, ladies, and commoners of Westeros?
It feels like an impossible claim to prove given that Eddard Stark is dead, Howland Reed seems to have vanished off the face of Planetos, and anyone else who was alive at the time with that knowledge appears to be dead and gone. EVEN IF Howland Reed manages to crawl out of the woodwork to corroborate Bran/3-eyed Raven's claim (assuming he even makes that claim) about Jon's true heritage, we're still just talking about the word of two relatively unknown (outside of the North) Westerosi noblemen.
The only way I can see of "proving" anything is if Jon ends up riding a dragon. But even then, would that, in addition to the theoretical claims of Howland Reed and Bran Stark be enough to prove that Jon Snow is actually Jon Targaryan (assuming R and L actually got married, and assuming that marriage is even recognized in the eyes of Westeros since polygamy is very much looked down upon...) ?
I am looking for theories or speculation based on the books, the TV show or historical events that end up with the people of Westeros generally accepting that Jon Snow is either:
Still a bastard of different heritage, or
A legitimate Targaryan, and trueborn heir of Rhaegar Targaryan.
Edit to explain first dupe:
I am not asking who knows, I am asking: how do you prove it in the eyes of everyone in the 7 kingdoms?
Edit based on S7E05:
In the scene where Sam decides he's had it with the silly Citadel, Gilly reads an old record of Rhaegar Targaryan receiving an annulment of his marriage to Elia Martell, and of his remarriage to... Lyanna Stark?
Assuming this is true, and assuming Jon Snow is indeed their offspring, then he is indeed a true-born Targaryan. Now as to how to prove it, obviously this record would come in handy when attempting to do so...
Edit to explain second dupe:
This is very similar to Is there any tangible proof of Jon Snow's lineage?. However, I am asking how it would be proved, either with tangible evidence or by proclamation or by some other strategy I hadn't thought of, in the eyes of Westeros. In other words, with proof, or no proof, how are the people (nobles+commoners) of Westeros to be convinced of this truth? I accept that this question is related, and should certainly be linked, but I don't think it is a duplicate.