King's estate, which is controlled by his sons, last year asked a judge to order their sister to surrender the items. In a board of directors meeting last year, Martin Luther King III and Dexter Scott King voted 2-1 against Bernice King to sell the two artifacts to a private buyer.
The two sides today told Fulton County Superior Court Judge Robert McBurney they are close to an agreement but not quite there.
A lawyer for Bernice King asked the judge to order mediation, and the estate's lawyer did not oppose that.
Both items had long been in Bernice's possession, and lawyers for the estate filed a lawsuit just over a week later asking a judge to order Bernice to surrender both items.
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Speaking from the pulpit of Ebenezer Baptist Church in Atlanta, where her father and grandfather preached, Bernice in February 2014 denounced her brother's intentions, saying the Bible and peace prize medal were among their father's most cherished possessions and shouldn't be sold.
The case was set to go to trial in February, but Fulton County Superior Court Judge Robert McBurney in late January temporarily halted all action at the parties' request to allow them time to settle. The judge ordered the parties to appear before him in late March if they hadn't reached an agreement by then. He later extended that deadline to Wednesday.