Yes, it's a false dichotomy. There are two requirements for a pair of choices to form a dichotomy, which are that they must be "mutually exclusive" and "jointly exhaustive". The former meaning that there must be no instances which are shared by both choices and the latter meaning that they must encompass all possible instances.

There are some people who would say yes for a different reason than what is given and same for people who would say no. Additionally, it's possible someone could think both things and still meaningfully answer one or the other (while ignoring the prescribed motive).