Although the rise in the Producer Price Index was greater than expected, most analysts agreed
that the index was unlikely to continue going up and that inflation remained essentially under control.
A. that the index was unlikely to continue going up and that inflation remained
This is correct and resolves all issues discussed below. First, the idiom "unlikely to" is correct. Additionally the second "that" in the second parallel makes the meaning clear and unambiguous. The analysts agreed THAT the index was unlikely to rise AND THAT inflation will remain the same. "xxx agreed THAT X is unlikely and THAT Y." If you don't include the second THAT, the inflation remaining the same would be unlikely, which is illogical to the rest of the sentence.
B. that it was unlikely for the index continuing to go up and for inflation to remain
This is incorrect because it poses a meaning issue. It was unlikely "for X and for Y." The logical meaning is that it was unlikely for the index to rise, but it was not unlikely that inflation was going to remain flat (it's the opposite). Also, the idiom is "unlikely to" not "unlikely for."
C. that the index was unlikely to continue to go up, with inflation to remain
This answer uses a "with" modifier, which modifies the clause before it. This has an ambiguous meaning. Does this mean that stable inflation makes it unlikely to see a rise in the index? Or is it the opposite? The true meaning of the sentence is that most analysts agreed on X and Y, not X, modified by Y. They are two separate things unless otherwise stated clearly.
D. on the unlikelihood that the index would continue going up and that inflation remained
Incorrect. This is a meaning issue. This answer is saying that there is an unlikelihood among analysts' thoughts that "x and y" will happen. The meaning of the sentence should be that it is unlikely the index will rise and that is inflation will remain the same. Both are not unlikely.
E. on the unlikelihood that the index would continue to go up and for inflation to remain
Incorrect. First, there is a parallelism issue as stated in answer choice D. Also, there is an idiom issue. The correct idiom is "unlikelyhood that" not "unlikelyhod for." The first component of the parallel structure uses this idiom correctly, but the second component, relating to inflation, uses the incorrect idiom.