And that simply makes sense, if they will meet (and why wouldn't they meet, if Gawyn goes to Caemlyn; except that it would be otherwise probably convenient for your theory...). Elayne knows that Rand didn't kill Morgase. Besides that, Egwene didn't tell Gawyn about Rahvin yet. Together with Elayne's argumentation, this will obviously change Gawyn's mind.

As said, it's pretty crazy to believe that Gawyn would nonetheless still risk the hate & death of the women he loves - plus, of course the destruction of the whole world - to avenge Morgase, only because he heard some rumors that Rand might have killed her; but all the other people he actually trusts most argue the contrary. Morgase isn't that important..

And after all, Gawyn also didn't even kill Rand in the end of LoC, when he had a good opportunity and was certainly even more convinced that Rand killed Morgase & still more hurt about her death than today.

It's possible that Gawyn and Rand will never become friends, but they certainly won't kill each other, if the plot is in any way realistic.