To Llewellyn's point, are you asking for synonyms or alternative means to communicate the concept?
Though admittedly verbose, it could be fun looking for a colorful way to explain the term in regards to the situation at hand.
"The Court finds that his external relationship to () provides the opportunity for bias, and thus has decided to disallow his ...
I think the closet words are dispute and difference of opinions. You could also try 'opposing interests' or 'competing interest'.
Maybe even cross-purposes.
Just giving multiple alternatives, because any of these could fix based on your context.