In one sentence Lukas et al., IEEE S&P 2023). Conclusion first: treat PII regurgitation as "reduce + audit + accept it can't be fully eliminated," and don't read "we ran scrubbing" as "no PII" — that's the most common false security here.
In one sentence synthetic data provides no anonymity by default. Stadler et al. (USENIX Security 2022) showed that any privacy evaluation based on "how similar real and synthetic records look" severely underestimates risk, and that synthetic data is no safer than traditional anonymisation — unless generated under a formal differential-privacy (DP) guarantee; and even with DP, you can't escape a hard privacy↔utility trade-off. Chen et al. (GAN-Leaks, CCS 2020) go further: even when the attacker can only sample from the generator (full black-box — exactly the "publish a synthetic dataset" case), membership inference (MIA) still distinguishes whether a given real record was in the training set above chance, and gets more accurate as overfitting rises. The only thing that gives a bound is formal DP, and ε is not zero.