In the abstract of that paper, it says "the significant negative correlation could be observed only in groups with above average mental abilities and not in a random sample from a general population." Are there other papers that say there's a general negative correlation?

Is that a universal correlation, or just looking at any particular area?

Because assuming people in a certain group have a certain level of skill, it makes sense that people with below average intelligence in that group may be compensating with extra conscientiousness, while people with above average intelligence for that group could have been part of a better group, but their lack of conscientiousness dragged them down.

I.e. being less conscientious doesn't make you more intelligent. Rather, the less conscientious you are, the more likely you are to end up surrounded by people with less intelligence than you (who make up for it with more conscientiousness than you).