They don't agree about anything!!!
Why is it called loggerhead anyway?
According to Oxford, it could be used as a weapon:
— PHRASES at loggerheads in irreconcilable dispute or disagreement. perhaps from a use of loggerhead in an obsolete sense long-handled iron instrument for heating liquids (when wielded as a weapon).
Can we consider a person who always wanted to further a discussion and asks lots of questions a loggerhead?
I wouldn't. ;-)