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So, unless there's information elsewhere, we don't even know if he left Middle-earth.
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I would say the phrasing of quote you gave, in particular the use of the word 'went', at least strongly suggests that Celeborn left Middle Earth eventually, even though the exact time is not recorded.
Perhaps Celeborn left on the last ship with Cirdan, since otherwise he would not have had 'the last living memory of the Elder Days in Middle-earth'.