U.S. Supreme Court Associate Justice Brett Kavanaugh attends his ceremonial swearing in in the East Room of the White House October 08, 2018 in Washington, DC.

Justice Brett Kavanaugh on Tuesday signaled a possible break with his fellow conservative Supreme Court justices during a dramatic day of oral argument in a high-stakes death penalty case.

The case concerns Russell Bucklew, who was convicted in Missouri on charges of murder, kidnapping and rape, and sentenced to be executed this year. Bucklew has argued that his execution, if carried out by lethal injection, would be cruel and unusual because he has a rare medical condition that would make the procedure excruciatingly painful.

In his questions to Missouri's attorney, Kavanaugh suggested he was sympathetic to that argument.

"Are you saying even if the method creates gruesome and brutal pain you can still do it because there's no alternative?" Kavanaugh asked John Sauer, the state's solicitor general.

Sauer responded that under precedent set by the high court it would be necessary for the person set to be executed to offer an alternative method that reduced the pain.

"So you're saying that even if the method imposes gruesome, brutal pain, you can still go forward?" Kavanaugh asked. "Is there any limit on that?"

Kavanaugh appeared to be testing the logic of a separate precedent established by the court in 2015. In Glossip v. Gross, No. 14-7955, the court ruled that death row inmates challenging the method of their execution must provide an alternative.