A lot of folks are incorrectly stating that "that" is incorrect or redundant after "one." While there are certainly forms that allow us to skip "that," it's not incorrect here. We can say "one that they believe" or "one that they believe," depending on whether our meaning seems clear enough without the "that." "One" is a pronoun here, but it works just the way a noun does, and we can modify it the way we'd modify any noun.
A few examples:
I shared a poem that I believed was masterfully written. (Here, "that" helps to make it immediately clear that I believed something
about the poem, rather than that I believed the poem itself. This is a common reason to include "that" or "who/whom.")
I shared several poems, including one that I believed was masterfully written.
I met many people that day, but there was one person I couldn't forget. ("That," or better yet, "whom," wouldn't be wrong here, but the meaning is 100% clear as is.)
Sitting across from me was a person I had believed to be dead.
Sitting across from me was a person whom I had believed was dead. (This second version feels clunkier. Although the "whom" makes it clear that I will be describing my belief, rather than saying I believed the person, the meaning was already fairly clear in the first version. I think some of the awkwardness comes from the switch from past perfect--had believed--to simple past--was. Although this switch is correct, it creates more work for the reader than the previous version, with its time-independent use of "to be.")
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