If Aragorn had carried the sword to Gondor, it most definitely have been recognized, and Denethor would have had proof to back up his suspicions. Also, if Aragorn had carried the sword while fighting the servants of Sauron, word would eventually have gotten back to Sauron. But, Sauron only knew that the Heir of Isildur lived when Aragorn revealed himself in the palantir at Helm's Deep.
Morwen's theory that Aragorn was carrying the shards of Narsil for a special purpose seems like a good one. But why would Aragorn leave his other sword at home? It can't be so hard to carry two swords that he would risk his life on it. Swords were the mainstay of the Dunedain. He could have worn the Shards of Narsil on his other hip. I suppose the only explanation is that Aragorn was a good enough warrior to not need a sword.
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