Can a singular antecedent use "their"?
Yes. Singular "they" and its possessive form "their" can refer to a person whose gender is unknown or irrelevant. A singular antecedent alone does not make "their" incorrect.
Correction: the former example "Each of the researchers submitted ___ notes" offered both "their" and "his or her." Both can complete that sentence, so it does not have a unique correct answer. It has been withdrawn from the practice questions below.
Source: College Board's classroom practice guide, page 77. Correction reviewed .