Unique NIZKs and Steganography Detection

Non-interactive zero-knowledge (NIZK) proofs tend to be randomized and there are many possible proofs for any fixed NP statement. Can we have NIZKs with only a single unique valid proof per statement? Such NIZKs are known under strong cryptographic assumptions (indistinguishability obfuscation), and are conversely known to require strong cryptographic assumptions (witness encryption). In this work, following Lepinski, Micali, and shelat (TCC ’05), we consider the following relaxed notion of unique NIZKs (UNIZKs):