At loggerheads with means in strong disagreement with someone. The correct answer is this meaning because the expression is used figuratively, not only by its literal words.

Examples:

• The union was at loggerheads with the management.
• In an HSC essay, At loggerheads with can express this meaning: in strong disagreement with someone.
• When this expression appears in a passage, connect At loggerheads with with the idea of in strong disagreement with someone.