Can you elaborate on what you mean by "later shown to be totally insufficient to support that conviction"? The sufficience of evidence doesn't randomly change.

Can you elaborate on what you mean by "later shown to be totally insufficient to support that conviction"? The sufficience of evidence doesn't randomly change. If you mean that they were convicted based on evidence that shouldn't have been shown to the jury, you can win an appeal on that. The standard before your convicted is that the jury must find you guilty beyond a reasonable doubt. The standard after you've been convicted is that you must have found substantial new evidence that warrants reconsidering the verdict, or you must show that the original trial was mishandled somehow. "I think the jury was stupid" is not a valid appeal.