Does a defendant have to waive his right to self-incrimination under both the Fifth and Sixth Amendments?

Defendant also contends that the court erred by not holding an in camera hearing on an offer of proof regarding inconsistent defenses. (He does not point us to anything in the record, however, that shows that the court would have understood his request as pertaining to an offer of proof regarding inconsistent defenses, if indeed it would have so pertained.) Citing Simmons v. United States (1968) 390 U.S. 377, 88 S.Ct. 967, 19 L.Ed.2d 1247, he maintains that he was unconstitutionally forced to waive one constitutional right to invoke another. He asserts that he had to "diminish his right to due process of law in presenting his basis for severance of the charges" in order to "protect the invasion of his right to counsel under the Sixth Amendment and his right against self-incrimination under the Fifth Amendment...."