The justices did not comment on Monday in leaving in place the death sentence for Charles Rhines.
Rhines was convicted in the stabbing death of a doughnut shop employee in Rapid City, South Dakota, in 1992. His appeal followed the high court's 2017 ruling that evidence of racial bias in the jury room allows a judge to consider setting aside a verdict.
Rhines claimed one juror said Rhines should not be sentenced to life in prison because he is gay and would be housed with other men.
The justices rejected a similar appeal from Rhines last year.
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