If Anarion lived past the War of the Last Alliance, would he have challenged Isildur's High Kingship and sundered Gondor from Arnor's control?

There is no doubt that Isildur would have committed Gondor's full and total kingship to his brother, because he did so with his Nephew Meneldil. But would Anarion do the same thing as his Son?

Meneldil took advantage of Arnor's relative inability to respond to his proclamation of independence because Valandil was a child. But what about Anarion? He has a stronger claim to self rule than his son and he won renown as one of the chief war captains under Elendil.

Also, if Anarion does claim sovereignty, what will happen to the Third Age in general? It's up to you if Isildur survives or not, you can paint the picture with either scenario.