As I understand it, under a grant of immunity for testimony, they're compelled to testify. They don't have a choice. They can't refuse the immunity either.
The only justification they could give for refusing to testify is avoiding self-incrimination. The grant of immunity removes that justification. They have to testify, whether they want to or not.
Not true, to my understanding. The immunized person can refuse and take their chances. Otherwise, the normal process of using the immunity carrot with the charges stick is bass ackwards.