I came across the below sentence :

Since the first successful organ transplant was performed in the United States in 1954, a serious problem faced by transplant recipients have been the possibilities that its own body will try to reject or destroy the transplant.

The only error pointed out is "have" that should be replaced by "has".
Shouldn't "possibilities" be "possibility"?

Should the sentence read as below as per your suggestions?
a serious problem faced by transplant recipients has been the possibility that their own bodies will try to reject or destroy the transplant
or
a serious problem faced by transplant recipients has been the possibilities that their own bodies will try to reject or destroy the transplant

I feel that the word "possibility" or "possibilities" should look upto "a serious problem" which is singular. As per the subject verb agreement, the first one should be correct. Please suggest.