A recent poll of elected officials suggests that candidates, when in the midst of a tough campaign, often make statements about an opponent that they may not think is true.
我个人觉得这个句子简化一下是这样的:candidates often make statements that they may not think is true.
其中that应该是跳跃修饰了。
从语意上来讲,candidates只能认为陈述是不是真的,不能认为一个人是不是真的
open to discuss~