I was not clear enough, thanks. Whether it is a pok or zk depends on the chosen signature scheme. In any case, zk signature schemes exist and are implied by the existence of one way functions and publicly verifiable nizk
Almost. The result is from CRYPTO89 paper of Bellare and Goldwasser. They derive a signature scheme from a nizk. It is not known whether you can get a nizk from a signature scheme. Moreover, no signature scheme can be a ZK: https://crypto.stackexchange.com/questions/35177/is-using-di...