The neither-nor pair is saying not this one and not that one, with one being the determiner for the verb usage. The non-standard use of they should actually be the pronound she since both names are obviously female.
I disagree- they could individually thinking about both passing.
When we say 'Neither-nor' we are talking of them as a whole; hence 'believe'. But if we say 'Either - or' we are talking of them as singular, hence it would be 'Either Jane or Sue believes they will pass.
In this case we have to apply the rule of proximity. When subjects in a sentence are linked by the correlative conjunctions "neither/nor," the subject that is closer to the verb determines the number of the verb. Thus, the correct answer is "believes".
jane and sue are plural so we use believe without "s"
Is it me or am I just going blind to see that believe sounds way better than believes or either. But i guess its just me.