It simply makes much more sense that Egwene or - at the latest - Elayne will finally convince Gawyn that Gaebril was the Forsaken Rahvin, that he manipulated Morgase and killed her, but certainly not Rand. That is after all, what these two women closest to Gawyn correctly believe (except for that Morgase is actually still alive, but they might also learn this soon). And if Egwene already told Mattin Stepaneos this story, it's only realistic that she would Gawyn tell the same now in one of their next talks, so that he learns about his mother's fate and buries his hate for Rand.

Besides that, it's at least as likely that Galad, Perrin & Morgase will already be in Caemlyn, when Gawyn might appear there, so that Gawyn sees that Morgase is alive.