One thing I find curious that may bear on this is Aragorn's statement to Frodo on their meeting at Bree.
Quote:
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'I did not intend to tell you all about myself at once. I had to study you first, and make sure of you. The Enemy has set traps for me before now.'
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That suggests that Sauron was worried enough about the idea that an Heir of Isildur remained to have in some way been looking for evidence of such a person. Aragorn, of course, does not elaborate on the "traps", but it says something that even before the War of the Ring the Heir of Isildur was on Sauron's mind.