What is the test for a writ of habeas corpus against a defendant who was not adequately represented by counsel at his state court trial?

He argues in the alternative that, regardless of whether he was adequately represented by counsel at his state court trial, the district court should have issued a writ of habeas corpus on the ground that he was legally insane at the time of the crime and his conviction therefore was in contravention of the due process clause of the Fourteenth Amendment. We agree with the district court's conclusion that petitioner did not sustain his burden of proof on this issue at the habeas corpus hearing. Johnson v. Zerbst, 304 U.S. 458, 468 (1938).¹⁵