*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](https://mathoverflow.net/a/22246/8133) 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](https://math.stackexchange.com/questions/3491349/projective-conservativity-of-choice), 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 $\mathsf{ZF}$.