Careless Singing Costs Lives

[Gordis]

It is hard to imagine that Sauron hadn't figured it out that at least Galadriel had an Elven Ring. He wasn't that stupid - and the choice was so obvious. He knew her personally (as Annatar), he knew Celebrimbor - how could the head of the Mirdain not give a Ring to Galadriel?

Then Lorien, with its closely guarded borders and its magic fences. I guess Amroth's realm (before Galadriel came there) had nothing of the kind. Must be quite obvious that Galadriel had a Ring.

And finally here is a quote from UT "the Hunt for the Ring:

But the Lord of Morgul, seeing no better counsel, determined still to seek northward, hoping maybe to come upon Gollum as well as to discover the Shire. That this would prove to be not far from the hated land of Lórien seemed to him not unlikely, if it was not indeed within the fences of Galadriel. But the power of the White Ring he would not defy, nor enter yet into Lórien.

Note how casually the magic fences and Galadriel's Nenya are mentioned - not a hint of doubt that this info was fully available in Mordor..