For example, abstract objects could be considered to exist in the minds of people imagining them, and consequently in some neuronal pattern, which may or may not match between different individuals, but considered to not exist as something independent of the conscious minds imagining them. While this is a version of nominalism, it is not nearly as clear-cut as "abstract objects do not exist".

That would be conceptualism and is a moderate anti-realist position about universals (if you're a physicalist). Nominalism and Platonism are two poles of a continuum about realism of universals. So, you probably lean towards nominalism if you're a physicalist and conceptualist.