To be in conflict. A "logger" is a 16th-century term for a block of wood, so a "loggerhead" is a blockhead or fool. They are at loggerheads over the best way to lead the committee.
if two people or groups are at loggerheads, they disagree strongly about something (often + with ) They're constantly at loggerheads with the farmers' union. (sometimes + over ) The Senate and the House are still at loggerheads over the most crucial parts of the bill.