Does a witness have to testify in a witness testimony under privilege against self-incrimination?

Further, a witness is not considered "compelled" to testify in the absence of claiming a privilege against self-incrimination. (Minnesota v. Murphy, supra, 465 U.S. at p. 427.) If a witness makes disclosures instead of claiming the privilege, the government has not compelled a witness to incriminate himself. (Ibid.) The high court has long "recognized that '[the] Constitution does not forbid the asking of criminative questions,' [citation], and nothing in our prior cases suggests that the incriminating nature of a question, by itself, excuses a timely assertion of the privilege. [Citation.]" (Id. at p. 428.)