AUSTIN -- The U.S. Supreme Court rejected an appeal without comment Monday for Brandon Daniel, who remains on death row for the shooting death of Austin police officer Jaime Padron.

Padron was shot and killed in 2012 after responding to the Walmart near Interstate Highway 35 and Parmer Lane on April 6 for reports of a shoplifter. Daniel struggled with Padron before shooting him in the neck. He was found guilty in February 2014 and sentenced to death.

Daniel's lawyers had argued that during the 2014 trial, District Judge Brenda Kennedy communicated improperly with jurors and refused to step aside when defense lawyers appealed that her actions were grounds for a new trial, according to a report from KVUE's news partners at the Austin American-Statesman.

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The report states the Texas Court of Criminal Appeals compounded the error when the criminal court declined to transfer Daniel's appeal to another judge, then accepted the district judge's recommendation that his appeal be denied.

Citing an affidavit from a juror, the Statesman reported Kennedy responded to a jury note without discussing or revealing what she said with Daniel's trial lawyers. Travis County prosecutors argued that there was "no competent evidence" of the alleged note or secret communications with the jury.

According to the report, Daniel's lawyers also argued that the defendant should be allowed to pursue a second claim that his trial lawyers acted defectively by missing evidence that suggested he was autistic -- a diagnosis that could have led jurors to choose a sentence of life in prison instead of a death sentence. Prosecutors also rejected this claim.