Earlier quoted context omitted.
Conscientious has more than one definition. Yes, that is one of them. I'm referring to the other definition which does involve right and wrong: https://www.dictionary.com/browse/conscientious
That is also the same. See the clarified example: "She's a conscientious judge, who does not let personal prejudices influence her decisions." Basically she does her duty as a judge instead of letting her personal feelings influence her judgements. Conscientious people do things by the book.
A judge not letting one’s feelings influence their decisions can also be a matter of conscience, because they believe that justice should be equal for all.
Conscientious people do generally go by the ‘book’ or convention, but it matters to them whether the ‘book’ is ethical or not.
That doesn’t mean they have a sense of universal ethics. You could certainly have a conscientious concentration camp commandant who believed that fascism was the most ethical system and that everything else was less honest, but the point is that conscience is part of the equation.