Yes, only "whichever" is possible there. "which(ever)" is required whenever the range of selection is restricted, and this is clearly the case because of the superlative (of all the opinions stated, there is one that has the most support):
- People in group situations tend to agree with whichever opinion they think has the most support.
"what" will only be accompanied by a noun in questions (direct or embedded). "whatever" may appear with a noun in any kind of sentence, albeit in a more open context, where there are no limitations as to the number of options presented for selection:
- People in stressful situations will tend to agree with whatever opinion is presented to them. (Notice that "what" cannot be used here for grammatical reasons, while "whichever" would not be suitable because the range of possible opinions is too wide.)