I just finished this test. These two different answers confused me, too. I chose A at first. But after checking the answers, I think "on condition that..."should be followed with a common clause. That is to say, it does not have to use the Subjunctive Mood which is used in A. According to the whole meaning of the sentence, the "on condition that..." clause should be in the Present Tense to express the common rules. So at present, I think C is the best answer.
Hopefully, more cders will discuss about it later!
636. Several senior officials spoke to the press on condition that they not be named in the story.
(A) that they not be named (B) that their names will not be used (C) that their names are not used (D) of not being named(A) (E) they will not be named