You cannot make self-serving statements under oath and then refuse to testify, asserting 5th Amendment protection.
Day v. Boston Edison Co., 150 F.R.D. 16, 21 (D.Mass. 1993):
(There is no question but that the privilege against self-incrimination can be waived, not only explicitly but also implicitly by failing to assert it. Thus if a witness who is compelled to testify,does, in fact, testify and during testimony reveals information instead of claiming the privilege,the witness can be said to have waived the privilege as to the information disclosed.")
Lerner made statements prior to invoking 5th Amendment privilege and can be compelled to testify on those statements.
I agree with your statement, but you really should try to find at least a U.S. Circuit Court citation.
Her statements were general and offered no substance for any specific line of questioning.
She can take the 5th on any or every specific question.