Rashi comments that although some maintain that he was the high priest, this opinion must be rejected since Zadok and Abiathar
held that office during the reigns of David and Solomon.
In response, Jesus asks if they remember what David did when Abiathar
was the high priest, how he went into the temple and ate the shewbread that only the priests are supposed to eat.
A supporter of Adonijah in his quest for David's throne, Abiathar
is one of the losers when Solomon eventually becomes king.
Archaeologists have determined that the biblical city of Anathoth to which the Levitical priest Abiathar
fled is Ras el-Karrubeh, but it did not survive the Iron Age (Peterson 1992).
Jesus' explanation seems a stretch (it was not even Abiathar
who was the priest in the 1 Samuel story he makes reference to).