Texas is preparing for the execution this week of an autistic man, Robert Roberson, who was convicted of killing his 2-year-old daughter in a case that has drawn intense scrutiny for its reliance on a questionable diagnosis of shaken baby syndrome.

A majority of the Republican-dominated Texas House has called for the execution to be halted. The detective who helped obtain the murder conviction now says he is “firmly convinced that Robert is an innocent man.”

Mr. Roberson would be the first person executed in a shaken baby case, his lawyers said. The diagnosis, a medical determination that abuse has caused serious or fatal head trauma, gained prominence more than three decades ago and led to a spate of criminal convictions. As evidence emerged that the diagnosis was not always reliable, some have since been reversed.

In Mr. Roberson’s case, the defense has insisted that no crime was committed at all. Lawyers have presented new evidence and expert testimony since the trial suggesting that Mr. Roberson’s daughter, Nikki, died in 2002 as a result of pneumonia and a prescribed medication she had taken.

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So far, the courts in Texas have been unmoved.

With an execution date set for Thursday, Mr. Roberson’s lawyers and his supporters, who include John Grisham, the novelist, have appealed to the state’s Board of Pardons and Paroles to recommend clemency, or a reprieve, and have urged Gov. Greg Abbott to grant it.

Mr. Roberson has sat for a television interview from prison. His supporters have made the case for clemency to Phil McGraw, the television host known as Dr. Phil.

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