Yes and no. Completion of high school is usually a requirement for admission to college, and so would just be presumed; one would only mention high school if one did not have any university-level academic achievement.
Would it be clearer if I said "post-secondary"? In US English, the words "college" and "university" are used interchangeably. Indeed, in conversation with an American I would have said "college level"; I only said "university level" because I thought that would make it clear to someone from outside the US what I meant; apparently I was mistaken!