This is sort of an anti-answer, which I've accordingly made CW, but here goes:

Whether $\mathsf{ZFC}$ is projectively conservative over $\mathsf{ZF}$ seems open; see Joel's answer from a while ago (and to my knowledge nothing has changed). Moreover, it is known that $\mathsf{ZFC}$ is projectively conservative over $\mathsf{ZF+DC}$ (see the argument here, which I think is folklore) and $\mathsf{ZFC}$ is $\Pi^1_4$ conservative over $\mathsf{ZF}$.

To the best of my knowledge, there are no candidates for a counterexample to projective conservativity here. Personally I have no guess about whether $\mathsf{ZFC}$ is after all projectively conservative over $\mathsf{ZF}$.