The man at the center of a Supreme Court case about posting threats on Facebook has caught the attention of U.S. probation officials for more recent comments he allegedly made about cross burning in a prison note to a prosecutor.

The high court heard oral argument Monday on whether to uphold the 2011 conviction of Pennsylvania’s Anthony Elonis for making threats on Facebook against his estranged wife, law enforcement officials and local elementary schools. The case is considered a test of free-speech rights in the age of the Internet.

Mr. Elonis, who said his posts were fictitious rap lyrics and not real threats, served a little more than three years in prison and was released earlier this year. Under the terms of his sentence, he remains under court-ordered supervision and must abide by an array of conditions, such as not using alcohol or possessing firea