A recent poll of elected officials suggests that candidates, when in

HI Sujit,

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.

Since your question is specific, let us dissect this sentence to know analyze the structure:

Cl. 1: A recent poll of elected officials suggests

Cl. 2: that candidates, when in the midst of a tough campaign, often make statements about an opponent

Cl. 3: that

Cl. 4: they may not think

Cont. of Cl. 3: is true.

(blue = subjects, pink = verbs)

Let’s focus only on the structure of the sentence. For a minute assume that Clause 4 “they may not think” is not there. So the relevant portion of the sentence now is as follows:
Candidates,…, often make statements about an opponent that is true.

As you can see now in this sentence there should not be any confusion that “that” is the subject of the verb – is. And obviously “that” refers to the noun phrase “statements about an opponent”. Now let’s bring back clause 4 – they may not think. This clause adds contextually important information - what candidates may think or not think about those statements. But it by itself has SV pair. So this clause definitely adds contextually important information but structurally it fits right in.

The subject of Cl. 3 “that” stands for “statements”. Since “statements” is a plural subject, we need a plural verb for that. That is why choice B (are) is the correct answer.

Choice A leads to SV number agreement error and the other choices fail to make a clause with subject “that” that refers to “statements”.

I hope this helps resolve the confusion. If you still have doubts about this then think of it the following questions - what is “that” associated with; what “is true”.

You can refer to OG 12#118 question as well. Following is the sentence with correct answer choice C:

The world wildlife fund has declared that global warming, a phenomenon that most scientists agree is caused by human beings's burning of fossil fuels will create havoc among migratory birds by altering the enviroment in ways harmful to their habitats.

Thanks
Shraddha