Which preposition to use with loggerhead

with Occurrences 10%

Perhaps if some fundamental truths were brought home to the people, or, to be more exact, to the peoples now at loggerheads with each other, a notion of the peril equally impending upon all concerned and the conviction that an indefinite prolongation of the present state of things is impossible, would prove decisive factors in restoring a spirit of peace and in reviving that spirit of solidarity which now appears spent or slumbering.

in Occurrences 1%

That was in flip days, when there were always two or three loggerheads in the fire.

on Occurrences 1%

Movements are not only duplicated, but reproduced a hundred times in miniature, in one denomination after another; special talent is restricted to a narrow field; buildings and church-plants are multiplied, but lie largely disused; sects and communities are at loggerheads on unessential points; all thisand the world is not being saved!

Which preposition to use with  loggerhead